My colleague Allison Benedikt has a worthy rant attempting to use moral suasion to persuade people not to send their children to private school. She's absolutely right. She also very reasonably says that private school should not be made illegal. Freedom, after all, counts for something.
That said for the public policy literalist in your life, I would say that the relevant issue here is taxes. Private elementary and high schools are, like many other classes of nonprofit institution in the United States, subject to some very favorable tax treatment. One part of this is that donations to private schools can be deducted from your income tax bill. For normal people, the charitable tax deduction isn't a particularly large subsidy. But for the kind of people who send their children to private schools and who pay very high marginal income tax rates, this can be extremely valuable. Second, non-profit institutions are generally exempted from property taxes which, again, can be a big deal in expensive cities.
I'm a little bit skeptical about both of these practices in general. But as applied to private schools it seems totally and obviously outrageous. A private high school may be a non-profit organization, but it's certainly not a charity. It's a private club for the benefit of the families involved. At best private school is a private consumption good like buying your kids expensive clothes or fancy toys. There's no reason municipal tax codes should encourage land to be used for private schools rather than houses or regular businesses and there's no reason the income tax code should encourage rich parents to spend money on private school tuition rather than anything else. John Cook's view that private school should be illegal goes too far, but I'm skeptical that hectoring alone is enough to solve this problem. Make prep schools start paying property taxes, and deny their donors lavish tax subsidies for their donations and I think we'll start to see some real change.