Private schools' advantage

Michael Gove's speech decrying the domination of the privately-educated has reignited an old debate. But there are two important distinctions to be made here.
One is: does private schooling confer advantages merely through the quality of its education, or does it confer them even beyond this?
If private schools merely offer better education, you'd expect their pupils to get more A levels and places at Oxbridge, but you wouldn't expect them to go on to succeed more than state school pupils with the same qualifications.
However, two pieces of evidence suggest they do better than this. One is that, as the Sutton Trust has shown, the top professions comprise more former private school children than one would expect from their roughly 50-50 share of Oxbridge students. The other is that men from independent schools earn almost 7% more (pdf) than men from social classes IV and V, even controlling for the type and quality of the degree they have.
This implies that you cannot eliminate the advantage that private schools have over state schools merely by eliminating their educational superiority - not that this is feasible, given how expensive it would be. Worrying about teacher quality and class sizes, as Gove does, is therefore not the whole story. Yes, it's a (big) part of the story. But not the whole of it.
The second question is: does private schooling actually cause these non-educational differences, or is it correlated with them?
What I mean is that they are likely to be due to things such as greater confidence, ambition, a sense of entitlement to top jobs and social networks.But it's quite possible that these would exist even if private schools did not, simply because they would be transmitted from rich parents to rich kids anyway.
What I'm saying here is that abolishing private education, as George Monbiot and Laurie Penny advocate, is not sufficient - whether it's desirable or not. The advantages of the rich would exist even if private schools were abolished. I agree with Shuggy: if you want to increase equality of opportunity and social mobility - and I'm not sure how many people do - you need to reduce economic and social inequality.
Another thing: I suspect that the disadvantage bright state-educated kids have against private school kids is smaller in the City than in the media - that Jim O'Neill is less atypical in finance than Paul Mason is at the BBC. Is this just me, or is there hard evidence for it?
Yet another thing: Given the prevalence of public school kids in acting, music and sport (other than football), I suspect an under-rated benefit of private schools is not so much their formal academic superiority, as the better extra-curricular education they offer.