Obviously, not all private schools are the same just as not all public schools are the same. It doesn't do much good to speak about them as if they are uniform. The biggest problem I see with private schools (aside from the 80% of them which involve religious indoctrination), is how they tend to siphon off children and parents who do care about education and would likely improve the public schools by being involved. Private schools have also enabled people of means to push for reduced taxes and funding of public schools without being concerned that it will have a direct negative impact on their children.

While individual private schools can be wonderful and highly beneficial for individual kids, I think there is sufficient data to show that on a macro scale, they're bad for society. If every parent who sent their kid to a private school used the extra commuting time and money spent on tuition to make their public schools better via advocacy and donations, I think it's likely that the public school situation would improve significantly.