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.
A lot of students would get much better educations in private schools than they could even dream of getting in public schools.
Of course, there would be a process of building new private schools, and there would be some that were, like public schools, of lesser quality at first.
Those would be weeded out by the market system in time.
Everything takes time.
No claim has been made that there would be ubiquitous perfection all at once.