Part of me wonders why old-fashioned private schools aren't good enough anymore. Most of the religious ones (and that's often a factor for people wanting to go this route) have pretty good financial aid, especially within the Catholic system.
You can stand anywhere in this borough , throw a rock and you will hit a private school.
This area is an extreme example because of free market forces not in play elsewhere, but it cost about $300/month/kid to send our kids. The Catholic system was more, but they field sports teams and such. (Sports teams that do pretty darn well against the public schools, I might add.)
A private education is not exactly out of reach for poor families if markets are allowed to work.