Pretty much everyone applying to Oxbridge is applying with top grades, plenty of exceptional candidates from both state and private sector schools get rejected.
If you want to understand the real issues in social mobility I suggest reading the Milburn report which I think is one of the best reports I've ever seen a government produce: http://www.bis.gov.uk/assets/biscore/corporate/migratedd/pub...
http://www.suttontrust.com/research/applications-offers-and-...
What private schools seem particularly good at is instilling a rather outgoing self-confidence that seems pretty uncommon amongst state educated kids. Note that this is effectively orthogonal to actual capabilities - but the end result is that I suspect that a lot of privately educated kids will come across a lot better in these situations than people from state schools.
It's not fair - but to be honest that's why you pay the considerable amounts of money for private education - to give your kids an advantage.
Assume for a moment that the exam results of students follow a bell shaped curve and that the mean and standard deviation of exam results in the state and private sector are the same, then you would expect the proportion of state school pupils accepted by Oxbridge to be the same as the proportion in the general school system. If instead we assume that the mean in the private sector is higher then you would no longer expect the same proportion to be accepted, and indeed that disproportionately many private sector students would get in.
I'm not saying this is the case (although private school students probably do do better in exams on average), but without knowing anything about the situation how can you accuse them of discrimination? The 'oh, I saw a really good person get turned away' argument is silly. Some colleges only offer a few places for some subjects but are presented with 20+ students who all have predicted straight As, good UCAS forms etc. It's inevitable that some good students will not get in, and it doesn't imply discrimination.
I read mathematics and my interviews were purely technical. (None of those "hundred uses of a toothbrush" for me - perhaps those were for the arts students ;-) )
In the UK we are in the unfortunate situation whereby school examinations are so easy (as a result of political intervention) that it is effectively impossible to distinguish between the best candidate on examination results alone.
Privately educated students do make up a much larger proportion of the student body than they "should", compared to the overall population. However, it could be (correctly) argued that state school pupils are more reluctant to apply in the first place.
I can confirm that at Cambridge between'97 and '00 there was a concerted effort by the university and the student body to encourage applications from statistically underrepresented sections of society.
Bla.
I do around 50-60 admissions interviews a year, and we don't even know to which schools the applicants went when we question them. What you've seen in action is a few anecdotes and whiny parents who blame external factors for their child's rejection letter.
NB, I'm not saying there isn't a problem. There's still a lot of room for improvement; both universities are doing plenty of outreach work and are fully aware of the situation. But your portrayal of Oxbridge is ridiculous.
With a technical subject, it's relatively straightforward to put a meritocratic admissions system in place. I don't think you can make that argument for humanities subjects.
I've seen this in action, exceptional state school students turned down from Oxford, and utter buffoons (but with the correct background and accent) being accepted.