Private education does not exist to be at odds with public education. It sexist for either profit, push particular ideology, or give an advantage. That’s not at odds with public education.

If there is no advantage, the profit motive goes as there is no advantage. Then all we are left with is religious schools. Most people using religious schools are not religious, but expect a good education where the public school is lacking. Improve the public schools fixed the problems and helps children. Banning private schools requires more funding for wealthier kids, so reduces funding for others, worsening education. Youre hoping that pressure from wealthy people would improve education funding. A wealthy vote and a poor vote are the same. So it is no more likely to be improved, while needing more funding to stay at parity of where we are now. You’re letting your ideology cloud the reality and outcomes.