You’re the one who thinks it’s good for everyone to be busy.
It’s also good to learn how to be happy in your own company and self-reliant. A different parent might prioritise that more.
I suggest you start with what each of your children actually enjoys and works backwards from there. It’s a right mission forcing them to do stuff they don’t want to do. Letting them do stuff they want to do is lovely.
I would not make Saturday school the deciding factor when choosing a school. I would think long and hard about whether it would suit my child.
If constant lifts would run you ragged and make your life unpleasant then that is, of course, also relevant factor. Your desire for them to be busy does not trump your own well-being.
As children get older they become more independent and can make their own way to more stuff. They also change their mind about what they enjoy and they also drop stuff. I’m not sure how old your children are. Are we talking junior or senior school?
You say you want them to be busy but you also don’t want to tire them out. Whichever school they go to, surely you can just strike the right balance between these two? Drop anything that is not enhancing their life.
I assume they’re sporty, because for a non- sporty child those Saturday afternoon matches are an enormous drag.