The Oxbridge programme is a bit of a red herring. Most schools put something in place. What they don't have is on tap expertise, funded support, specialised careers, Oxbridge and UCAS advisers. Often the trumpeted London based state schools are highly selective (this filtering out other issues), highly funded , and receive extra funds from interested organisations and charities. Grammar schools have an able cohort so are more likely to devote time to HE applications and also do this more efficiently because of streamlining of aspirations. From what I read on here, many. private schools have at least one member of staff paid to work with high fliers. Careers advisers in state schools are often shared, and not always very au fait with university stuff.
In any school, SLT set priorities, based often on what the governors ahve agreed and Ofsted set targets. Any member of staff who comes along and says they need time/ funding etc to set up a special bespoke Oxbridge group above and beyond a few meetings that already exist and could they please leave the school site with another member of staff and take some students out would, sadly, be ignored/ told to go away. That said, it does mean those who have got into Oxbridge have truly done well.
I don't know what the answer to your questions are either mid. I do think a goal needs to be access to HE full stop for some young people - and raising the status and profile of every institution so that institution snobbery became less of an issue (instead of suggesting that deprived young people should just jolly well see that going away to uni is a spiffing good idea when they can't afford it). Pipe dream in England perhaps. Any degree increases potential earnings, especially for BAME women. The Sutton Trust , for a range of reasons (some quite political) has got a bit tangled up in elite universities of late and may be missing wider points about participation and, oddly, not focusing on costs of uni much.
I think it will definitely be interesting to see the outcomes of the LMH foundation years at Oxford and the nascent scheme at Cambridge too.
The fact that a bunch of fairly well off, middling ability kids at my schol may not get the world's best advice and may end up at , I don't know, Staffs instead of Swansea is probably a small fry issue. Arguably , even the fact that some might end up at Portsmouth or NTU instead of Liverpool or Kent also really doesn't matter that much in the grand scheme of things. It's just quite frustrating when on MN to see how much support and help some independent school students get as a default : I do sometimes wonder what happens to some of them at uni thought when all the helicoptering vanishes.
Parenting does also play a huge part in all of this!
As LBJ once said 'you've got 1 problem; I've got 1001'.