I don't know which way you'd make more. I was thinking it would be a pain to do things individually and to work within the school system. If your an outsider, they can't just hand out contact information.
Then there might be an issue insurance. You might have to rent out the teaching space. If they do it for one private teacher, they probably have to open up to all music teachers. If they let one kid out for a private music lesson, why aren't they letting other kids out for other individual things.
And a criminal background check.
It's probably a good idea to get permission from the school board. If you walk in with your plan, your own background check in hand, and can get them a piano for their school, that might interest them. If there are lots of parents who want that, they will listen. If parents want it, they will find a way to make it happen.
I guess I was thinking it would be easier to get paid by the school than mess with individual payments and parents. Because the parents aren't going to be there at all I would think. And for resume appearances, it looks better to be hired through an organization rather than just doing things yourself.
I'd be concerned about who's guarding the piano though. I wouldn't want to be responsible for it at all if it's left at a school. The kids are hard enough on them, but I've seen adults swing the school piano into walls when they need it moved.