Perhaps if I speak more personally. I went to a high-status mixed college -at a time when the majority of colleges were single sex - within a high-status university. All the people who supervised me in my chosen subject were white independently educated men.
A significant proportion of the student body were independently educated and they, I think, found college life a little easier to negotiate. The college was run rather like a progressive boarding school. (The Senior Tutor was an old Etonian.)
Anyway for whatever reason it tended to be a higher proportion of those who were independently educated who went on to do research degrees at the same university and/or gain Fellowships there.
I absolutely accept that academic life has become increasingly international. (It's also clear from my visits to see my daughter, who's at a similar nearby place, that the student body is more diverse.) So I'm sure that there are many academics from top American universities there now, as well as others from similarly elite institutions all over the world.
Nonetheless if I look at my old College there are still an astonishing number of a) staff who were young academics when I was there and b)fellow students from my time there who are now academics.
I also have academic friends in Russell Group universities who have said that their own student and academic background makes it unlikely they'd be offered an Oxford or Cambridge Fellowship.
If institutions can be international yet - simultaneously - insular, I think it's likely that Oxford and Cambridge manage to combine those two qualities.
Does this affect how candidates from a variety of backgrounds are perceived at interview? Hard to say...