You're assuming private schools wouldn't be around to solely make as much profit as possible. Plus, there's always the issue of higher pricing being associated with better results. And then the schools could even follow the path of universities and start charging more and more.

It’s non-linear for most of those things though - a class of 20 kids or a class of 30 kids still need a teacher (although there’s evidence that the former’s going to have better outcomes), books aren’t bought new every year so are amortised over a whole load of students, lunches are cheaper when you can buy ingredients in bulk etc.

Private schools are usually more expensive because there’s a lower ratio of students to teachers. If all the rich kids left, you’d have schools being merged together to save money; you wouldn’t have the few(er) poor kids left behind with one teacher covering multiple classes.