If a public school fails, well, that's just too bad. If a private school fails at educating its students, it soon goes out of business. If a public school fails, it gets a bigger budget. Public schools have only to promise they will do their best. Private schools have to give parents and students what they want.
Public schools can give parents and students whatever they, the bureaucrats, want.
Private schools have to really educate their students. Public schools just have to take them in and see them out. Moreover, private schools have to compete against public schools that offer education for free. And that's not all. Private schools, and even home schooling, can only exist by government permission.