Oxford & Cambridge are very selective and entry is very competitive. Unusually for UK universities they hold a separate entrance exam for all courses. (Information about the extra admissions tests done by some universities and some courses: https://www.ucas.com/applying/before-you-apply/what-and-wher....)
Fun fact: When I applied to Cambridge I was interviewed by Mary Beard.
In maths, if you do well enough in the exam, you have to really mess up the interview to not get in though I would say. If you're borderline, the interview is important though - the interviews will usually be 95% maths questions so it's more like doing an oral exam where you have to explain your thinking process and answers.
I'm sure it probably depends on the discipline (and indeed the college, since some are harder to get into than others). I can see how this would be important in mathematics where it is hard to avoid rules and structure. In other areas, the decisions will be much more subjective.
Any elite selection process has high risk for not selecting the best people.
Either due to corruption or, even more likely, institutional biases.
Malcolm Gladwell has a few episodes of revisionist history that looks at some US universities who fall into this trap.
Even with all that said; the chances of a selection process for 17/18 year olds being able to correctly select the best future adults in their field is very low.
Another good example; the number of young geniuses or prodigies (whether maths, chess, acting etc.) who don't make anything of themselves.
Having interviewed maths applicants for Oxford, I don't think it's just corruption or biases - there are just more really bright students that you have places for and have to reject some people! We all tried really hard not be biased but I don't think there is any way to guarantee that you aren't somehow, however hard you try.
It's very hard to predict from a half hour interview precisely who will do well in future, and in fact some people might do better having gone to other universities. Oxford and Cambridge aren't the best choice for everybody - I'm sure being a bigger fish in a smaller pond suits some people better.
No judgement here. And you are right btw; we are biased constantly in everything we do, and it's a better person than you our me that can truly dodge it!
I read a study a while ago (sorry Google didn't find it again) that said a lot of exceptionally gifted people just don't apply to elite universities. Instead, the main characteristic of applicants tends to be that they are extremely driven.
Drive tends to not correlate well to long term success without talent as a companion. Therefore a lot of people at Cambridge (which I think was the example) tend to do no better than the average in later life (notably wasn't true for PPE grads obviously)
Did Malcolm Gladwell ever address the lack of scientific rigor in his work, or does he still roll with whatever narrative he thinks will entertain and sell?
Fun fact: When I applied to Cambridge I was interviewed by Mary Beard.