Increased demand for private schools will result in an increased supply of private schools.
Of what quality. To meet the demand, they will have to hire public school teachers and public school administrators. Or the bottom half of the educations school graduates from the Universities. The point will to be to make money, not necessarily provide a high quality education.
I support private schools. All my kids went to Catholic parochial schools. The schools cost me an arm and a leg, they did an excellent job of educating my kids. But you are not going to educate the 50 or so million kids of school age in private schools that are the equivalent of the one my kids attended.