Ehh, I mostly agree with your first point but I think it's kind of a pipe dream to think it could be otherwise. Go visit a public school in a place like Greenwich, CT and then go visit a public school in the Bronx. Both technically public schools, but students have a clear advantage in one vs the other.
Sure, Bill Gates's kids aren't going to public school probably even in Greenwich. Still, let's not pretend that the elite only go to private schools, or that they wouldn't congregate in wealthy areas to make public schools essentially private.
Inequalities will always exist as long as people can use their assets to gain advantages for themselves/their families, and trying to force it otherwise against an unwilling populace is likely to just exacerbate the problem in other ways (see: Suburban flight).