Depends very much on the course, I think, and the university. I was the only kid from my high school applying to Cambridge, and what prep they gave me was quite misguided, but I don't think it's possible to fake the process (in-person, test with ten hard problems (not requiring any knowledge outside the standard high school course, just more problem-solving ability) where you're not expected to solve more than two or three, then discussing them with your interviewers). I felt like there was some of what you describe with the people I met in humanities courses (of course you can attribute some of that to STEM snobbery), but on the side I saw, once you applied (and sure, I suspect there are people with the ability but without the wealth or connections who are too intimidated to apply), they wanted the best, and I'm sure they got them.
I got a place after my state school simply advised me not to bother applying. Independent schools don't give such misguided advice.
33% of Cambridge students (today) come from the 7% independent sector, with privilege evident at every level: independent candidates far more likely to apply, independent candidates still more likely to be offered a place upon application.
I don't have the data about the results but I suspect this trend reverses.
