This is a little off subject, but I am not sure it puts you at a huge disadvantage, but I agree there is some. My wife went to Vandy, her sister went to Vandy, her brother went to Wash U, Princeton, and just moved from being a prof at Penn to Columbia. I went to Pepperdine after my PhD. All of us are products of public schools.
I do think there could be a correlative relationship, but not necessarily a causal one on higher acceptance rates from private schools. If your parents care enough (and can afford...which my family could not) to pay for your education at a private school...that means there is a high probability that you will share their belief in the importance of education. Because 100% (or very close) of those people are paying, you would expect a higher percentage of those who value education vs a school that is free with your tax dollars. That leads to higher success rates for those from those schools, which feeds into the whole thing.
Anyway, maybe its just my field but I work with lots of folks from the ivies and MIT. Some of the folks are impressive and some aren't. Just like having 5 stars in recruiting is a good indicator that the athlete in question will be successful; being from one of the top schools is also....however we all know that there are a lot of 5 star busts as well.
Sorry for the ramble.