Back to the OP's question - if you read the current research, the correlation between success in school is more related to the amount of parental involvement than the level (or type) of school.
One interesting story -- my 1st grader's teacher has developed "literacy bags" that go home weekly with a book to read and questions, games, and activities all related. It is extra and not required, but the kids love them! We were sitting at his sister's sports practice this year and a teacher (also waiting) was amazed and impressed with the bags and how much time and effort they took from the teacher as well as the weekly commitment by the parents to sit and do them. The ironic part ? My son is a public school kid and she is a private school teacher. Her comment, "most private school parents don't want to be involved and they believe they pay for us to do all the work with their kids". Do I believe that is true for everyone ? Absolutely not! But it was her comment, not mine - and she has much more experience with the private school parents than I do.
You can generalize and justify anything you want. For us, $10k wouldn't "buy" a private school education worth anything - and it would actually be worse than the local public school system. I admit, I am fortunate to have an excellent school district.
Carrying that on to college... those lowly state schools (the college equivalent of public school) were found to be the three most desirable from a hiring perspective according to a WSJ study last year. In fact, 19 of the top 25 were state schools. We can all come up with our reasons why, but it was eye opening for those who believe that the Ivies are the key to long-term success.