Coffee this may all be about age.
In my day, showing how old I am, if you did not get into Oxbridge you probably went to Bristol or Durham, as an "Oxbridge reject". I was unusual in that despite going to a private school I did not apply to Oxbridge, and rejected Durham, because I had decided I wanted to be in London. Warwick, York, UEA, Kent, Southampton, Bath and Sussex were somehow lumped together as "new Universities" and a definate tier below. In the middle you had red-bricks like Liverpool, Manchester (including UMIST) and Birmingham.
When I got to LSE we were still busy being revolting and occupying the Director's Office. Staff, who believed in Milton Friedman, were far more right wing than the students. And us private school kids found ourselves very much in a minority surrounded by lots of bright state schools kids who did not have the support to do that extra term and sit Oxbridge exams. (My naice girls school use to forbid pupils from applying though I seemed to be exempted on account of my rebellious streak.)
Lots has changed. Bristol is no longer seen as the place where those who narrowly miss Oxbridge go. LSE (and UCL and Imperial) are, though northern comprehenisve kids are now an endangered species and students, who all seem to want to be merchant bankers, are probably more right-wing than staff. Warwick, Bath, Exeter and York all seem to have improved their offer. Bristol is seen as a good University, and a great place to live, but no longer seems to hold the status it once had.
DCs school was odd in that it was very cosmopolitan and so decisions were probably more closely aligned to decisions of ambitious overseas students. It appears though that lots of Private school pupils are still influenced by their parents or are attracted by the city, whilst Bristol in parallel are working hard on outreach and offering lots of contextual offers, presumably to help create a more balanced student community.
I am still surprised that it is not seen as desirable as it once was. And wonder why this is.