From the outset, I am not a scientist, and have read a bit about sugar and diet to try and understand things better. What follows is my understanding about sugar. There may be some mistakes here, but I'll try and reference information where I can.
It largely depends on what you define as "deprive".
If you completely remove all sugars from a child's diet (lactose, fructose, glucose, sucrose etc.), then yes, this is largely going to be detrimental to the child since the human body does require sugars (there's also evidence to suggest that removing sugar from your diet and relying on a high fat diet can, paradoxically, lead to insulin resistance).
Digging deeper into the types of sugars out there, watch Dr. Lustig's video "Sugar: The Bitter Truth". While there is some controversy around what he says, from what I've read I do believe there is a large dosage of truth in what he describes. I will try summarise what I can recall.
Glucose is not very sweet, and tends to be better processed as a sugar by the body, primarily due to the fact that 80% of glucose can be broken and used by most of the organs in your body. Every cell in your body can pretty much use glucose. 20% of the calories from glucose hit the liver, gets stored as glycogen, which the liver can store in pretty much massive quantities without ill-effect, and be used as energy (Lactose gets broken down by an enzyme called lactese, but I won't get into that here, since this is not really relevant.) It's a lot more involved, as it also sends good signals to the rest of you body to increase insulin, tell you to stop eating etc. etc. A pretty good loop that is healthy.
Fructose and sucrose, both of which are infinitely sweeter, however, gets broken down by the liver, roughly 80% of the calories. Unlike glucose, cells and other organs in your body cannot break it down on their own.
However, this is only part of the story. Bacteria in your gut can also break down fructose, which is why fructose from fruit (that often come with high levels of fibre) has less of a health issue than added fructose in low-fibre foods, since the fibre causes the sugars to stay in your gut longer, hence the bacteria there can deal with it better.
Added sugars that are ingested with very low levels of fibre (think fruit juices, soft drinks, very sugary foods like cakes etc.) tend to go straight to the liver to get broken down in almost the same way that ethanol is broken down, which can help lead to increased levels of obesity and health problems if consumed in large quantities. There are a number of other factors at play, such as large quantities of added sugar inhibiting signals from your body telling you that you're full and had enough to eat, and that you're no longer hungry.
What Dr. Lustig proposes is that fructose and sucrose are essentially a poison in the way your body deals with and processes them. Many of the health effects linked to high consumption of added fructose and sucrose are similar to health issues seen in alcoholics.
The actual process is a lot more involved than the brief description here, but is described at length in Lustig's video.
All this suggests that restricting added sugars in your child's diet is a very, very good thing.
In your particular case, one thing to keep in mind is that people are different, even children from the same parents. Experiences, drugs, food etc. all affect people in different ways, and some people derive intense pleasure from things that others don't. All children respond to stimuli in a different way, and from what you describe, your child derives intense pleasure from things that are sweet. Sugar is addictive because it is a type of hyperpalatables, that is, a foodstuff that stimulates the pleasure centres of the brain. Processed sucrose and fructose, high fructose corn syrup etc. are intensely sweet, far more than lactose and glucose, and thus trigger much higher levels of pleasure than other types of sugar.
Some people are going to be affected by this more than others. Children in particular are very susceptible to experiences that trigger their pleasure sensors because it can be very intense for them. Think about it: it's a new experience, and if it's pleasurable, they want to repeat it. They haven't developed the willpower to recognise good things in moderation. If I were to just think of my child, for example, he gets great pleasure from very particular types of activities (for example, watching a TV show), and if I remove those activities, it can lead to behaviour such as you describe: tantrums, jealousy at others who are allowed to take part in this activity, longing etc.
It seems to me, then, that your child is likely not trying to satisfy a dietary need, but is rather trying to satisfy a pleasure craving. I'm not saying it's not impossible that he has a requirement, but it does seem highly unlikely since he doesn't seem to show any other health issues you'd relate to a lack of something in his diet. If you are worried about it, then seek professional advice, but I think it is highly unlikely.
As hard as it may be, I'd suggest you're on the right track by pushing through and teaching him to control his impulses to satisfy this pleasure craving. Satisfying this craving with sugars from fruit is definitely a good solution. Rewarding his tantrum with the very thing he is craving (that is, processed sugars in sweets, soft drinks etc.) is going to cause more problems for you later on since he's going to repeat this behaviour.
