"Westminster is clearly great at getting kids to apply, but then doesn't seem to surpass others schools spectacularly in its acceptance rate, in spite of its hugely selective nature. You would expect some kind of parity."
I don't quite follow this. I would expect a school which is especially good at getting kids to apply to have a lower acceptance rate, not a higher one (because kids from that school with relatively lower ability will be applying, whereas kids with that same level of ability from a school which isn't good at getting people to apply won't be.)
I don't think we can possibly know if universities are biased too far one way or the other. If they don't take into consideration that some kids have had much less suitable teaching, parents whose kids don't get in from state school comps will scream "bias towards private schools!" If they do take it into consideration, parents whose kids don't get in from expensive selective schools will scream "negative bias for kids who have worse results than mine!"
I imagine Molio works in admissions. When DH was marking entrance papers, he could have told you, for schools which had lots of applicants, which tended to get more in too.
FWIW, I got in to Oxford, many years ago, from a state open entry sixth form. I'm quite sure that there were kids from academically selective private schools who got better A levels than me but didn't get in. But then I was the second person from my school to take Further Maths, and the first to pass it (there was one other in my year who also failed).
I was about average ability at Oxford. I guess the question would be whether the people of lowest ability there (or at least lowest achievement at the end of the course) there tend to come from top academic private schools, less academic private schools, grammars, comps, sixth form colleges...or whether it's a mixture roughly matching the mixture at intake. If it's a mixture, the selection process is probably about right.
That's a statistic I'd be very interested in.