The only time there has been social mobility was in the 50s-80s, with a decent grammar school system in place, bolstered by grants. There’s also the fact that university in now incredibly expensive for your average lower middle-class family.
Oxford has also since opened the gates to privileged international students, leading to a Swiss finishing school air about the place, especially amongst the postgrads.
Public school children generally adapt to Oxbridge easier at the start because the system is familiar – very work hard, play hard, in a 24/7 institutional environment. I could pick out the state school kids a mile off in freshers week, either trying too hard or just shrinking in the corner.
In terms of admissions, prospective students need an adviser who understands the colleges, and can match the student to the college they are most likely to fit in.
But is it worth it overall? I spent seven years at Oxford and looking back there were some really low times, and I expect I would have dropped out if I were paying the current tuition fees.