This is a really interesting nuanced thread and has reminded me of the importance of not jumping to conclusions.
Very interesting to read @cantabsupervisor insight. I went to Oxford many moons ago from a really crappy private school (large classes, all ability, teachers without qualifications) and I could see the difference between my education and my many friends who'd been to Westminster. Those doing English A level spent the whole of lower sixth/y12 reading the canon and only started their A level texts in y13. I was so far behind and was actually asked to leave. I caught up and ended up with a first but I felt pretty demoralised throughout. So I'd agree that a minority of private schools churn out very Oxbridge-ready pupils.
But where the nuance comes in response to Trinity Hall is that it's for three subjects alone. And these are, bar a few very niche subjects like Anglo-Saxon, by far the least competitive subjects to get into but also the ones that bright ambitious state school kids are really unlikely to want to study. I work with these pupils and they all want to do STEM or economics or medicine. I suspect that these subjects could take 99% of their students from state schools and not have a drop in quality.
What Trinity Hall are trying to do, as far as I can see, is shore up these dwindling departments. As PP says - they want to persuade a smart Oxbridge-ready kid from St Pauls Girls to apply for classics at Cambridge instead of economics. I know a girl who's at Oxford doing classics - her college made four offers. Only two of the pupils got the grades to take up their places. They're desperate to widen participation in these subjects but they don't want students who, if they can't get 3As at A level without strong extenuating circumstances, are unlikely to do well.
The whole Oxbridge superiority thing is totally diluted now due to the enormous gulf in selectivity between subjects. You can't compare getting an offer for modern languages with computer science. A tutor told me that they will make an offer to all state school boys applying for French or German as they're such unicorns. Of course they still have to get the grades and I'm sure are worthy it's incomparable to competing for a place to do engineering.