Huh my understanding is the opposite!
Almost everyone in the US seems to go to public school. Nobody seems to talk about where they went to school - they just went to where was by their home. For example this list of privileged people at Beverly Hills High School is extraordinary
https://en.wikipedia.org/wiki/Beverly_Hills_High_School#Nota...
The US public school system seems to be doing extraordinarily well here - in almost every other country these rich people would have been in a private school wouldn't they? The US system seems unusually egalitarian.
Most homeschooling parents do a variety of things, including teaching in groups with other homeschoolers, and hiring tutors for specialized subjects etc.
If the goal of the ban on private schools as suggested is to force the rich and powerful to send their children to public schools so that they are incentivized to make them better, then you would need to also ban homeschooling because otherwise those people could still school their children privately in cooperation with other rich and powerful parents via the homeschooling model.
I don’t quite understand how restricting a private school of several hundred students affects a parent teaching their children at home.
Clearly rules could be in place to separate those concerns apart.