Now I might have a false impression of what private school is like and what it creates, but I will never not find it incredibly wrong that people can pay for a better education and give their kids a perceived head start in life just because they can afford to.
I think we're agreed, that academic achievement is highly likely to be better. And in that I include artistic and sporting pursuits.
I guess they are handicapped in terms of mixing with 'normal' people, but advanced in mixing with stockbrokers, CEOs , politicians.
If your children want to be doctors for example though , while a state school isn't stopping them, their chances of 'success' are higher in a private school.
--edit-- I'm not saying this is right, in fact it's obviously wrong. But I'm teasing out what you mean by perceived advantage.
--edit edit-- I guess the less savoury elements of public schools, are primed to be in positions of power and influence , so we see them more (yes Boris, I'm looking at you) and form more negative views of them.
(come to think of it, the doctors we (as in my family ) see frequently , certainly sound privately educated, but throughly decent people)