The idea that we should create a ban on private primary and secondary-ed institutions to fix the public system is incomprehensible to me. Should we force people with enough time and effort to devote to quit homeschooling, too? How about those who can afford private tutors? How far do you plan to take this to eliminate any economic advantage? Inciting conflict and anti-governmental feelings is a great way to inspire creative workarounds by the upper class, not creative solutions to a broken shitpile of an educational system. Ask yourself: would you really be inclined to fix something that someone else broke because you were forced to by the government?

The solution to fixing a broken system isn't to prohibit an alternative and force everyone to use it; those with resources will find a way to circumvent any specific ban, as that's significantly easier than solving a systemic failure. Also, why is it their problem? I'm by no means a libertarian, or particularly anti-government at all, but why should individuals with higher socioeconomic status be burdened with repairing a malfunctioning piece of government above and beyond voting and paying taxes?

What the US public educational system needs is to incentivize the most qualified individuals—potentially those who graduated from private secondary schools, even—to join and help fix the system as it currently stands. The vast majority of teachers in US public schools aren't there because the pay and benefits are any good, and I've known several former educators who've since moved on to other careers due to the need for a living wage. If we want to restore the system correctly, it needs to be done by providing impetus for its repair, at which point perhaps those individuals will actually admire it enough to send their own future children there.

The real issue isn't private schools, the wage gap, or any of that, it's the dismal lack of motivation for anyone capable of overhauling the system to do so.