I would disagree with you, blackomen. There are private schools and there are private schools. I wouldn't spend money on a cheap christian private school, which is just a public school with slightly fewer poor kids. Those seem to be common across the country. But elite prep schools? You bet your ass. You don't get anywhere near that sort of education anywhere, including from parental involvement, unless you happen to have advanced degrees in multiple unrelated fields... you overestimate how much direct parental involvement in education matters once you get to the high school level, and underestimate how much peers matter.
On pretty much any metric you would choose to name, apart from getting laid, the best kids from public high schools were far below the best from the hoity-toity fancy-pants private high schools, where I grew up. Sports, competitions, even standardized tests. And absolutely, definitely, academics.
People here like to talk about how great a deal public college is compared to private. They forget two things. One, the best private colleges are essentially free, despite their high sticker price, unless you have enough money that the cost isn't a crisis for you. In that alone, it's clear that they win on cost. Second, it is often true that freshman courses are hardly different in material covered between private and public, with the exception that the good professors teach the same material in a deeper and more rigorous way. However, public schools are far behind on the depth offered in later years and in graduate programs. If your choice is between a meh public school and a meh private school, obviously pick public. If you have the opportunity to learn at the best, go for the best. The best has better people there; the inertia carries that forward.
I was lucky enough to attend one of those stereotypical new england prep schools. I was the token poor kid; yay, scholarships. After that school, college was a breeze. You wouldn't believe the difference in what's offered, and what's expected, between a school like that and the local high school. Now, for elementary school and maybe middle school, I don't see the point - in fact, probably better to go public and not be so sheltered and naive. But when you want to transition from day care (elementary) to education...
Vacations are fleeting, education is forever.