It is certainly the case that Public Schools in the United States could be better. With proper folks in charge of the funding of teachers and school facilities they can improve. A for profit approach is not part of making public schools better.
Also, it is certainly the case that, some private schools, offer very high levels of education to those that can afford it. But this does not in any way translate to the public school system being in some sort of "... race to the bottom ..."
Because expensive private are good has absolutely nothing to do with with public schools except in the case that some folks who are able to and also do send their kids to private schools have in some cases a greater hand in the funding of public schools and tend toward such nonsense as for profit schools.