I would consider jellyfin + ersatztv. ersatztv lets you create "live" channels and define your own programming. Instead of the kids having a free for all of being any to stream any of the media you give them access to, you instead give them access to a few channels.
One of the nice things about it, is the channel can go "offline" at certain times. So, if you have a strict bedtime of 8pm, the shows will literally finish at 8pm, and the channel will stop. No more "let me watch on more video!"
I have several channels like:
- Cartoons
- Education
- Sing alongs
- History
that they can switch between, but that is all I give them.