Also, this argument that students at super-selective independents are 'Tim but dim' types is only trotted out by people who have literally no idea. Obviously, independent schools, like state schools, are not all the same - far from it!
Admission to certain London schools at 11-plus is insane -more competitive than Oxbridge. For some schools, 20 children may be competing via exams and interviews at 11 plus for one place. All of these kids will be among the most academically able at their primary schools. These schools can literally cherry-pick those on a clear trajectory to all 9s and A stars. They will take the ones who are not only the most academic, but who do music, sports or 'something else' to high standards alongside their academics - simply because they can.
Once in these schools, high standards and competition are normal. You could be in the lowest set for maths, believe 'maths is not my thing,' and still come out with a 9 - as my DS did!
DS applied to Cambridge with 10 x 9s at GCSE and 4 A stars predicted and was rejected. He applied again with the 4 A star A levels achieved and got in. He found the work load and expectations at Cambridge easy compared to the workload and expectations his school. At school, he always felt very average - never got a prize or was never stand out in any way. Not surprising really as students with 9s and A stars were not particularly celebrated, just the normal standard expectation.
At Cambridge, DS was in the choir, rowed and very involved in about 3 societies. He was also out most nights, from what I could make out. He came out with a First with Distinction (80% overall and some of his exams were in the 90% range). If he hadn't reapplied, he might have always thought he wasn't up to the mysterious 'Cambridge standard' but actually it's a degree, like any other and they will make if it what they make of it if given the chance.
I would say the vast majority of kids from his school could have got Firsts from Cambridge without breaking too much of a sweat had they applied / got in. A large proportion applied to the US though and went that route. This was a few years ago and still seems to be a growing trend.
It's the same with the super-selective grammars in London where there are 20 applicants per place. Of course more of these kids are going to be applying to Oxbridge than they are from comps and of course they will be less phased by the workload and expectations once there.
DS may have been from an independent school but he is first generation uni and not white / British, unlike the vast majority of 'state school' white British at the uni. The point being, you can't have a chip in your shoulder about any of these things, just apply and do your own thing as best you can. He went to Imperial after Cambridge which is frankly much more 'elitist' (in an international 'elite' sense)!