This will only work if you also forbid rich people from homeschooling (= private tutors supplemented with sports & social clubs full of other rich kids—and very-well-funded rich-parent homeschooling support groups would probably start to look
an awful lot like private schools themselves) and from sending their kids out of the country for school.
Day school rates for good private schools (there are way, way more bad private schools than good ones) are in like the $20k-$60k/yr range. That's a lot of money to put toward avoiding public school, even if US private schools themselves are outlawed. You'd have to also mandate public school attendance, no alternatives whatsoever.
You'd also have to do something to prevent rich people from effectively buying whole school districts and turning them into private schools. There are already districts kinda like this—unleash the entire upper-middle and upper class on the current public school system, and pretty soon there will be a few dozen districts nationwide where it's impossible to buy a house for under a million dollars and the schools may as well be private schools.
