This is a common misperception held by private school parents, in my view. They see lots of kids from private schools go to Ivy League colleges and their equivalents. But it is not the private schools that accomplish that feat. It is the parents themselves, usually by virtue of the fact that they are Ivy League grads.
The legacy advantage is big. If one of your parents attended Yale, that gives you a leg up. One parent I know made careful notes about all of the students at her daughter's very prestigious private school who got into her very selective college. Most of them had parents who attended that same college.
I haven't seen any data to back this up. It is one of those elitist things that the colleges and private schools don't like to talk about, but it seems right to me. Anybody have any information to the contrary?
We do have data showing that all things being equal, a student is LESS likely to get into a selective college from a prestigious private school than she would if she were attending a regular public school. The less competition for Ivy League spaces at her high school, the better off she is.