A title of nobility that is granted. I do not know about Harvard, but Oxford which I will assume is not so much different, has most of its graduates from private schools, especially Eton college seems to be their favoured. These kids, due to having the badge of Oxford are quite favoured in the highest ranks of law, politics, etc. I kind of wonder how much different is the teaching in oxford from any other university and if it is so different then why do other universities not adopt their system.
Very few people from other backgrounds go to oxford, therefore it is quite a good analogy i'd suggest.
Around half of UK-national Oxford students come from state schools (http://www.timesonline.co.uk/tol/news/uk/article4027010.ece), and as that article says, that's at least in part because teachers scare their kids away from applying because of the very perception you're espousing.
What's more, independent schools are a very varied lot. The vast majority of the independent sector is day schools which are effectively high-end grammar schools with a strict dress code. I'm not going to pretend there isn't an advantage to going to one, but they're not substantially better than the best schools in the state sector: that's borne out by the statistics.
The teaching in Oxford and Cambridge is much more intense, and has a much greater emphasis on one-to-one and one-to-two teaching than any other Britihs university. That's where the real advantage lies; the reason it's not adopted elsewhere is that it's eyewateringly expensive. (The odd legal status of Oxford and Cambridge, historically, was at least in part to support the cost of the tuition system).
The problem with Oxford and Cambridge is one of perception rather than reality. The kids who get the grades (think 1600 SAT for our American friends), and who apply, have a decent chance of getting in.
