Non-profit/not for profit schools can work just as well as for profit schools. The key is to open up the market and have the businesses fight for customers. I probably said it before, but my wife is a teacher at a private, non profit school. You hit the nail on the head whey you said there is no incentive to do well in public schools. At my wife's school, teachers don't get tenure and aren't part of the union. The kids have to try and perform or they are kicked out. "Management" cares because if grades drop, tuition will drop as well. Everyone is on board. Even the parents. The vast majority of parents care how the kid is doing because they're paying for it.
I wish charter schools were around when I was a kid. I got so bored during some of my classes. I actually failed chemistry class because I was so bored. I got 50s in the class (mostly due to homework grades) and got something like a 90 on the Regents test. The teacher was good, but I had no incentive to work hard. I would pay attention in class, and then not bother with the homework.