Have been involved in Oxbridge admissions in the past.
IME, what we tended to be wondering at interview (assuming that the candidate was talented enough and self-motivated enough - that's checked but pretty much assumed) was "how would it be to spend an hour or two hours a week in the company of this person, with one other student, talking about the topic they were sent off to research and write an essay on?"
So we were absolutely looking for independent thought, a preference for learning through conversation rather than having information fed at them to digest and critique alone later, the ability to defend a position and to change their mind mid tack - to think aloud and on the hoof and coherently, and to be a generous conversationalist.
There were people I interviewed who were certainly bright enough but would have been emotionally crucified, in my judgement, by spending an hour or two hours a week in intensive academic discussion with me. Some of those we were able to pass on to colleges where the tutor was the kind of personality which would be nourishing to them.
Some people were so bumptious that the prospect of spending an hour or two hours a week with them was just unbearable. Some of those we were able to pass on to colleges whose tutors enjoy the company of that sort of exuberant confidence.
Some people were very attractive to us, but there simply wasn't room for them, and they fell through the net because, having applied to an oversubscribed college, the places at the less fashionable colleges tended to have been filled by the time they filtered down.
And from my current position, I have to say that those of us at the other Russell group universities are DELIGHTED to pick up the Oxbridge rejects, they tend to have a LOVELY time with us, get good degrees, be good intellectual company, get plenty of intellectual stimulation outside Oxbridge and go on to have happy and successful lives. Especially the ones who shrink into their shells at interview - because the lecture plus careful pastoral attention model often suits them to a tee and they blossom and flourish into splendid splendid intellects.