He has time off in the school holidays, he wants to go away in that time off and he expects you to find childcare so he can go away - either you take time off, or they go to your parents (assuming you have to ask them), or you find kids clubs for the 10 year old and trust the 12 year old to be ok on their own for the day
Why is any of that a problem? He might be a parent but that doesn't mean he's never allowed some time out for himself.
I would presume that if a parent isn't working in the school holidays and the other is, its the non-working parents job to look after the children, or to sort an alternative if they want to do something else.
He's listed a whole load of solutions but OP isn't happy with any of them. Why does she get the final say?
If you use up your holiday allowance for the summer, how does he plan to cover October half term, Christmas, and February half term?
They pay for childcare, just like everyone else does? The 12yo doesn't need formal childcare anyway so it's just the 10yo really. They can do a combination of holiday clubs, staying home with their older sibling, going to friends' houses, seeing their grandparents and some time with their parents. Isn't that normal?
OP is in a hugely privileged position to have a partner who can take the entire summer break off work to look after the children. Begrudging him three nights away is just ridiculous.