I think interviews are a good thing, for Oxford as well as students.
The process advocated by TiP is the one that DS went through when he applied for a Masters at Oxford. A paper application, with a chance to suggest what college he might prefer. Then a departmental acceptance, with a much later confirmation of college. (He got the college he rather randomly requested, but it was a reasonably obscure one.)
This all happened without seeing the University or the University seeing him. Yes by this stage it will be reasonably clear which applicants are up to the course, and he was more than able to look at the course content and decide whether it suited. However I think it would have been useful for him to see the place, meet some staff, and get a better sense of the learning environment he would be in for the next two years.
You get the same at UG level, though not at Oxbridge. Lots of kids don't get a chance to go to Open Days. Instead you fill in your UCAS form and if all goes well you get an offer. Acceptances at the end of the cycle (which both DC had) are too late for offer days. UCL used to, and maybe still does, hand out offers to UK based students at specific offer days. Part marketing presumably, but it will help prevent students turning up without a clear idea of what they are letting themselves in for. Important in an Oxbridge context, plus with luck you will meet people who become your fellow students and have some familiar faces on your first day.
I also think interviews give the chance to probe more and to discover raw talent. By all means review interview training to reduce unconscious bias, but perceiving a problem does not mean the solution is a radical change.
(I agree with Sendsummer that a natural bias will be for students who interviewers want to teach, and why not. A disappointment for DS in his first year in London was that it was not the done thing to say anything in classes, though this changed as courses got smaller and they got to know each other better. Selecting students who will contribute cannot be bad.)
My best guess is that, for oversubscibed courses at least, there is a lot more wrong with sifting by PS than by interview. You cannot get a consultant to interview for you! And aside from STEP my understanding is that performance on aptitude tests does not correlate that well with subsequent University performance. Certainly UKCAT, the medicine aptitude test, appears more like the old Tiffin 11+, where the more you practice the better you will do.