The private schools are seeking and selling quality. Plus, the majority of private schools are non-profits. Many are religiously affiliated and would like a large population as as a potential source of new religious members, but they also need some way to fund that. If a family is already paying $5k/yr in property taxes it is less likely they can afford another $10/yr for 2 kids to go to private school. The funding is the biggest blocker to increased private and charter school attendance. That's why there's so much talk about things like vouchers, 529s, etc.

"Finally, why can't we just fix public schools?"

That's a good question without a good answer. Perhaps the best answer is the meta that people don't agree on the course of action that would "fix" the schools since there are numerous opinions on what that is, resulting in no action or half measures.