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tl;dr

Don't focus on the fact why they were hurt and that they deserve the pain. Lose this negative conotation. Focus on the solving the actual problem - your kid has their hand injured. Your attitude should improve. At least your negative bias will move slightly to the neutral. Jokingly speaking, if you are less hostile you are more sympathetic :).

And you are not the one to change their behaviour; your wife shall think about consequences of her doing too.

I think there are several issues there and you are not the only one to be blamed for them.

1. You and your wife take opposite actions when something bad happens.
   You do not cooperate rather anticipate the partner's actions. This is both confusing for the kid, they may think that if they follow your rule, they will go against your wife's and vice versa, and misleading, they can be taught how to "divide and rule" you in the position where they wanted you.
   Have a calm and factual discussion how and why you will treat your kid. Accept the resulting approach that shall be somewhere in between your and your wife's one. Stick to this approach and if one of you fail, remind it when the kid is out off vocal range.
2. Your kid runs in ransoms frequently.
   They do it because they get the attention they wanted when on ransom. This part will be tough to anticipate, they must know that the ransom doesn't lead to any change at all but them becoming tired. And I also think that this way they are abusing the issue #1 for their advantage.

Your point, that the lesson shall be taught, is a good one. But the way how you try to acomplish it fails. You focus too much on the cause of the accident and conclude that they deserve the pain. I think if you defocus from the failure/wrongdoing you also may lose the antipathy you are concerned about. You will lose the negative bias that prevents you to feel their pain.
Try to focus on the solution of the problem - in your case the injured hand. Ignore, laugh off or mock the accident itself. Show them that the injury is not fatal at all, even if it is serious. Your actions, on the other hand, must be serious and appropriate. Also, if you mock their bruised hand, you shall mock and laugh off your bruised hand too.

Your wife's point is also good. The cold antipathy is not good. But too much sympathy is not good either. When the kid is comforted only there is no lesson taught. Also your wife shows them that she cares more about them than you. If she blame you afterwards in front of them, I hope she do not, she teaches them not to respect you at all.

The goals of both of you ("Everyone shall face consequences of their doings" and "Our relationships shall be friendly and empathetic.") are good and they should, and can, be accomplished. Do not trade them off. Trade off you ways how you try to achieve them.

Once again: Take your kid to one of your parents' for a while. Go sit down and calmly discuss all your and your wife concerns just between four eyes, without any witness. Describe calmly your point of view and listen to hers. Find a way that both of you accepts; the one where the lessons are taught (your key target) the positive and empathic way (your wife's target). As a side effect the ransoms may fade away slightly or even completely; your relationships with both your kid and your wife will improve; your kid will see you both as equal authorities and friendly accomplices.

Next time anything unforseen/bad happens to your kid, DON'T PANIC! Focus on the result (humiliate the cause reference and the ruinig "I told you" commment) and stick to the checklist you agreed on with your wife. If you fail to stay in rails, re-rail yourself as soon as possible and be the one who initiates the discussion afterwards (when the kid cannot watch you). If your wife fails, give her time to re-rail as well and give her the time to initiate the discussion to. Show the respect to your significant other. If anyone of you fail to notice the derail, be polite and stick to the facts. Do not blame. Do not bring any bad emotion at all and be sorry if you do so.

Kids know that it hurts (and it does hurt like nothing ever did before) but they do not have any idea how serious it actually is (They have # years of expirience and you are ## years in advance). They learn the scale and how to react by your response.

• If you laugh it off saying "What a bang!" you teach them that it is not at all serious if they fall down, got hit by doors etc. Be sure that you laugh off your own falls and hits too!
• If you comment it "It was your fault", "I told you (you idiot)" or "Have taught your lesson" you teach them that any failure near you is a no-no and that you are completely unhelpful and unfriendly guy. Someone one has to endure for as short ammount of time as possible.
• If you start theatrical "Aw, what a horrible accident. Does it hurt? Does [whatever] help?" Whenever they start to cry and eventually burst in ransom you teach them only one thing: Crying and ransoms lead to attention to ME and comforting ME regardless what actually happened.

Kids are not scaled-down adults. But they learn really fast. Freakingly fast, to be honest.

  If you continue by commenting the bruise or bulge in funny yet positive matter, like "Wow, that one is growing nice" you:

• redirect your focus from the cause of the accident and your urge to comment it. You also discontinue the reference point for you "They deserve that" reasoning for your antipathy.
• redirect the kid's attention from the pain.
• Focus the attention of both of you to something that has nothing in common with both pain and blame - laugh.
• can examine the injury and assure yourself that nothing really happened.

After the examination there are options:

• If nothing actually happened and the pain is already fading propose returning to the game or proposing new one.
• If it is not serious and it doesn't seem to fade fast comfort the injured part focusing on relieving the cause of the pain and ask whether the pain is fading. Focus on the fact that the pain is fading and it will disappear (in short ammount of time).
• If it is light wound play a funny and friendly Doctor for them with fast cure and encourage them to face the pain and that it will fade away soon (And you will play again).
• If it is serious, take them to hospital. Encourage them to face the pain and that the doc will help.

Allways keep good attitude to what's happening (Do forget why it happened!) and be positive to the future.

I know several small kids that fall down in very spectacular accident no profesional stunstman could ever survive and the kid jump and run away laughing at it. Some even with stunning comment "shit happens". It is pleasure to have such kids around and I believe they have really good relatiosnhips with their parents too.

And as a bonus for you, the kid will remeber the fun with you, even the accident mocking parts as a funny time with you. And if you focus on the solution (like healing the wound) without reminding them their fault (I told you) you will set yourself as a pal to go to with a (serous) problem.

Shorter answer to title question only:

As anongoodnurse noted, you seem to prefer rational thinking over the empathetic one and I copy it. On the other hand I think if you change your rationale little bit you don actually need to switch from pure rationale driven to pure empaty driven.

I think your issue starts from the very first reference point you take in such situation: the cause of the accident. Maybe the more you are blamed the more you hate the cause making the first reference point stronger and the positive feedback loop do its job. 

Your following logic seems reasonable: Wrongdoing implies punishment; Not obeying your order implies brused hand. This doesn't deserve any sympathy, does it?

Consider focusing on something else; covering the first and negative emotion with a different, less negative or even positive one. This starting attitude affects whether you would feel sympathy or antipathy with the injured kid. I think if you focus more on solving the injury; whether it is serious, what was actually injured, etc.

This focus covers the negative emotion in your head opening the way for the more sympathetic you (the one who feels the sypathy); it brings your actual and visible attention to your actual kid, not the kids fault, so the more sypathetic you is percieved (the one who displays the sympathy).

Sooner or later you should do this with fewer and fewer rationalisation and it may become something automatic. Maybe you would feel it.

Consider also laugh the injury off, when you carefully examined it. This way you, again, cover and defuse the negative emotion and replace it with something positive - laugh. You also teach the kid not to focus on the pain rather to focus on something joyfull.
This laughing off puts another emotional bond between you and the kid; another way for your more sympathetic you to come in. This bond stronger your overall sympathy to them and it also show, or discover, this sympathy to them.

Try to focus on fun with your kid and belittle their troubles. The rest should come on its own.

I also think that the rational you should still be near the wheel; in case of emergency the emotionless person (no fear, no shame, no mercy) is the best one - they focus on the crucial tasks first. Then the empathetic you should come in and heal the wounds the rational one did. The rational one saves lives; the empathetic one helps them heal.

Slightly longer answer addressing all the exclamation marks that have risen in my mind:

I think there are several issues and you are not the only one to be blamed for them.

1. You and your wife take opposite actions when something bad happens.
   This is confusing for the kid, they do not know what is right and what is wrong. All they know is that if they do A they will be rewarded only by you but if they do B they will be rewarded by your wife.
   This also teach them that if they do ~A they will annoy you but it will be tolerated by your wife and she will back them up in case of possible punishment. They will learn how to fire countermeasures so the punishment focused on them will be deflected on one of you.
2. Your kid runs in ransoms frequently.
   They do it because they get the attention they wanted when on ransom. They are comforted (by your wife, mostly).
   There is also another scenario - and you wrote it down in the question already.
   
   You told them not to stay behind the door and they ignored it. (A wrongdoing that deserves punishment)
   They are hit with the door and injured.
   You tell "I warned you not to stand here." (You deliver a punishment)
   A ransom starts. (Countermeasure fired)
   Your wife breaks in and starts comforting. (Punishment deflected) You are blamed for the injury and the ransom (Deflected punishment backfires to you).
   
   You can see how you 3 year old kid effectively exploits the inconsistency in your and your wife's behaviour.

Your point, that the lesson shall be taught, is a good one. Everyone shall take consequences of their doing. The earlier they learn it, the better for them and the less painfull lesson it will be. On the other hand, the way how you try to acomplish it is not accepted by your wife - your last and only ally.
I think you focus too much on the cause of the accident and rationally conclude that they deserve the pain. I think if you defocus from the failure/wrongdoing you also may lose the antipathy you are concerned about. You will lose the negative bias and your rational deduction may lead to completely different attitude, certainly the more sympathetic one. 

Your wife's point is also good. The cold antipathy is not good, people need empathic response sometimes. But I think too much sympathy is not good either. When the kid is comforted only there is no lesson taught. Also your wife shows them that she cares more about them than you, in other words using examples above, they are taught that the successfully deflected punishment will strike you harder that it would strike them. If she blame you afterwards in front of them, I hope she do not, she teaches them not to respect you at all.

The goals of both of you ("Everyone shall face consequences of their doings" and "Our relationships shall be friendly and empathetic.") are good and they should, and can, be accomplished. Do not trade them off under any circumstances.

Try to calmly discuss all your and your wife concerns just between four eyes, without any witness. If you describe calmly your point of view and listen to hers you will find what she finds wrong, and why she finds it wrong, and you will have a chance to explain what you find wrong in her approach.
This way you can find a way that both of you accepts. 

As a side effect you can have more healthy relationship with your wife, there shouldn't be any blames if you do what you both have agreed on, should they?
Another side effect can be in the total change of the battlefield rules; Once you fire a punishment (whatever it is) they will find their countermeasures (ransom) are no more effective. If your wife would back you up they will also learn, that countermeasures no more deflect the hit but rather amplify it.

And you are not the one to change their behaviour; your wife shall shink about consequences of her doing too.

And you are not the one to change their behaviour; your wife shall think about consequences of her doing too.

I think both approaches, your "I told you not to stand there." and your wife's comforting are bad ones.

1. You and your wife take opposite actions when something bad happens. You do not cooperate rather anticipate the partner's actions.
2. Your reaction is completely cold and repulsive. There is no empathy at all. You are teaching your kid not to go to you when something really serous happens. They will expect unhelpfull and annoying "I told you" reaction.
3. Your wife, on the other hand, teaches your kid that if they start crying they will get her attention and comforting. There is no lesson taught but it could, and should, be taught. A bit too extreme, but the point is clear: Everybody "loves" such a kid.

Kids know that it hurts (and it hurts like nothing ever before) but they do not have any idea how serious it actually is (They have # years of expirience, you are ## years in advance). They learn the scale by your response. If you laugh it off saying "What a bang! " you teach them that it is not at all serious if they fall down, got hit by doors etc. Be sure that you laugh off your own falls and hits too!

If you continue by commenting the bruise or bulge in funny yet positive matter, like "Wow, that one is growing nice" you can examine it and assure nothing really happened.

If you want or need to comfort them, comfort the injured part (bulge, bruised hand) not "them" (you poor little Timmy). Do not ask wheter it is hurting (it certailny is), ask whether it is fading. Allways focus on the better future, not the painfull present or past.

If it is serious (broken bone, bleeding,...)laugh the accident off but be seroius regarding the consequences; go to hospital if needed and mock the accident when describing what happened. "They broke the doors with the head and only this happened to them." Introduce your kid to the doctor as a brave hero, not a whimper.

tl;dr

Don't focus on the fact why they were hurt and that they deserve the pain. Lose this negative conotation. Focus on the solving the actual problem - your kid has their hand injured. Your attitude should improve. At least your negative bias will move slightly to the neutral. Jokingly speaking, if you are less hostile you are more sympathetic :).

And you are not the one to change their behaviour; your wife shall shink about consequences of her doing too.

I think there are several issues there and you are not the only one to be blamed for them.

1. You and your wife take opposite actions when something bad happens.
   You do not cooperate rather anticipate the partner's actions. This is both confusing for the kid, they may think that if they follow your rule, they will go against your wife's and vice versa, and misleading, they can be taught how to "divide and rule" you in the position where they wanted you.
   Have a calm and factual discussion how and why you will treat your kid. Accept the resulting approach that shall be somewhere in between your and your wife's one. Stick to this approach and if one of you fail, remind it when the kid is out off vocal range.
2. Your kid runs in ransoms frequently.
   They do it because they get the attention they wanted when on ransom. This part will be tough to anticipate, they must know that the ransom doesn't lead to any change at all but them becoming tired. And I also think that this way they are abusing the issue #1 for their advantage.

Your point, that the lesson shall be taught, is a good one. But the way how you try to acomplish it fails. You focus too much on the cause of the accident and conclude that they deserve the pain. I think if you defocus from the failure/wrongdoing you also may lose the antipathy you are concerned about. You will lose the negative bias that prevents you to feel their pain.
Try to focus on the solution of the problem - in your case the injured hand. Ignore, laugh off or mock the accident itself. Show them that the injury is not fatal at all, even if it is serious. Your actions, on the other hand, must be serious and appropriate. Also, if you mock their bruised hand, you shall mock and laugh off your bruised hand too.

Your wife's point is also good. The cold antipathy is not good. But too much sympathy is not good either. When the kid is comforted only there is no lesson taught. Also your wife shows them that she cares more about them than you. If she blame you afterwards in front of them, I hope she do not, she teaches them not to respect you at all.

The goals of both of you ("Everyone shall face consequences of their doings" and "Our relationships shall be friendly and empathetic.") are good and they should, and can, be accomplished. Do not trade them off. Trade off you ways how you try to achieve them.

Once again: Take your kid to one of your parents' for a while. Go sit down and calmly discuss all your and your wife concerns just between four eyes, without any witness. Describe calmly your point of view and listen to hers. Find a way that both of you accepts; the one where the lessons are taught (your key target) the positive and empathic way (your wife's target). As a side effect the ransoms may fade away slightly or even completely; your relationships with both your kid and your wife will improve; your kid will see you both as equal authorities and friendly accomplices.

Next time anything unforseen/bad happens to your kid, DON'T PANIC! Focus on the result (humiliate the cause reference and the ruinig "I told you" commment) and stick to the checklist you agreed on with your wife. If you fail to stay in rails, re-rail yourself as soon as possible and be the one who initiates the discussion afterwards (when the kid cannot watch you). If your wife fails, give her time to re-rail as well and give her the time to initiate the discussion to. Show the respect to your significant other. If anyone of you fail to notice the derail, be polite and stick to the facts. Do not blame. Do not bring any bad emotion at all and be sorry if you do so.

Kids know that it hurts (and it does hurt like nothing ever did before) but they do not have any idea how serious it actually is (They have # years of expirience and you are ## years in advance). They learn the scale and how to react by your response.

• If you laugh it off saying "What a bang!" you teach them that it is not at all serious if they fall down, got hit by doors etc. Be sure that you laugh off your own falls and hits too!
• If you comment it "It was your fault", "I told you (you idiot)" or "Have taught your lesson" you teach them that any failure near you is a no-no and that you are completely unhelpful and unfriendly guy. Someone one has to endure for as short ammount of time as possible.
• If you start theatrical "Aw, what a horrible accident. Does it hurt? Does [whatever] help?" Whenever they start to cry and eventually burst in ransom you teach them only one thing: Crying and ransoms lead to attention to ME and comforting ME regardless what actually happened.

Kids are not scaled-down adults. But they learn really fast. Freakingly fast, to be honest.

If you continue by commenting the bruise or bulge in funny yet positive matter, like "Wow, that one is growing nice" you:

• redirect your focus from the cause of the accident and your urge to comment it. You also discontinue the reference point for you "They deserve that" reasoning for your antipathy.
• redirect the kid's attention from the pain.
• Focus the attention of both of you to something that has nothing in common with both pain and blame - laugh.
• can examine the injury and assure yourself that nothing really happened.

After the examination there are options:

• If nothing actually happened and the pain is already fading propose returning to the game or proposing new one.
• If it is not serious and it doesn't seem to fade fast comfort the injured part focusing on relieving the cause of the pain and ask whether the pain is fading. Focus on the fact that the pain is fading and it will disappear (in short ammount of time).
• If it is light wound play a funny and friendly Doctor for them with fast cure and encourage them to face the pain and that it will fade away soon (And you will play again).
• If it is serious, take them to hospital. Encourage them to face the pain and that the doc will help.

Allways keep good attitude to what's happening (Do forget why it happened!) and be positive to the future.