Children should be involved in, and have input on, life changing things - but while they get an opinion that should be considered they do not get a vote; that is why they are children not adults.
You are the parent(s), "man-up" and make the hard adult choices, while considering their opinion and wishes, but more importantly what is best for them.
Nothing is scarier for a child that the supposed grown-ups not being in charge and not having a clue. Once they are ready to have a vote and responsibility on important stuff they are adults. 16-19 years depending on where you live.
The degree of involvement on life changing stuff they have input in depends on the child more than age, but as long as it is input, not choice, it is better to give too much than too little.
Divorce however is a bad time for this, and may be exception to this rule.
The more important issue is, that as described, one or more parents are trying to weaponize the kids against the other. DO NOT DO THIS!
If not now, later your kids will hate you both for this.
This is no win nuclear scenario.
If you are an adult, act like one, and do not pull the kids into your fight.
How did it feel as a kid when your "friend" made you choose between your friends?
How much worse do you think it will be if a child / young adult / actual adult has to choose between their parents?
I am pretty sure there is a place in heck for parents who makes their kids choose between them.
How your kids will be affected by being made pawns in your fight, and how they will view the two of you when older, makes arguments about schools and activities utterly unimportant!
Even if one of you end up having no legal input, and punishingly limited time to spend with your child, do NOT cut child out of your life in spite. The child will feel you abandoned them - and you did. The child is unlikely to forgive you, nor should they. Your pride, hurt, and anger made you abandon them just because you could not have your way on what school and what activities they had. For shame on both parents if they let this happen!
It is more important the children can and are allowed to love both parents than what school or activities they do (even of one or both parents may be "unlovable" by objective measures*). The children will have enough anger for both parents for getting divorced to go around anyway.
I am a single parent, and a child of divorce.
This is my experience:
My mom could have told me (true) stuff as a child / teen, and turned me against my dad - she didn't and I am grateful. *The things I learned about my dad later in my 30s would have justified her doing so - but I am better off because she took the high road.
My children were not made to choose between my ex and I, we agreed on that much if nothing else.
My Ex did say a lot of negative things to our two kids about me over the years. I choose not to.
One kid accepted it at first and hated me for a while, but later realized the truth (at about 16 or so), and now hate my Ex and refuse to see or spend time with my Ex. I feel sorry for both their loss. I never asked the child to choose, but when they did as young adult at 16 I respected it (legally kids can override divorce settlements as young adults at 16 where we live).
The other kid, bless their heart, refuses to be placed in the middle and calls/ed out anyone who tries to do so. I regret the stress carried by kid because of it.
Both young adult kids have later thanked me that I have not vented my resentments of Ex to them.
If none of above makes you think of the children, know this:
The first thing my youngest child, just 4 at the time, said when we told him mommy and daddy was getting divorced, and would not be living together anymore, was:
"But who will take care of us? Where will we live?".
This was a scared kid who just wanted to know s/he would be OK after the grown ups blew up their world. S/he was not in a place to make choices about staying with mommy or daddy and live with those choices later. At least my Ex and I could say "We both love you! You will take turns living with both of us, and you will be OK with two places to live. It is not your fault mommy and daddy are getting divorced."
I hope the two of you find a way to protect your children from yourselves. Divorce is hard enough, don't turn yourself and each other into monsters, for the sake of your children.
(Apologies for long post and awkward wording trying to keep this gender neutral)