Far better to get a qualified teacher in that subject or someone who is a subject specialist if you're looking at GCSE + level.
I respectfully disagree there. First of all, there aren't many qualified classroom teachers who also want to do private tutoring, because they're busy enough already. Second, the appeal for many families is that I teach a range of humanities subjects and am NOT a specialist. So if their child struggles with, for example, essay structures in general, it's an advantage that I can tutor English AND French AND RS and work on the same issue in all of those subjects with their child. I've been employed as a sort of "general humanities coach" by more than one family.
some parents thinking that if a tutor has been to St. Paul's etc., they must know how to get in
I can't comment on this issue with regards to schools, but with regards to Oxbridge interview preparation (which is a large proportion of my work) I can completely see why they want a tutor who has been to Oxbridge. The point of Oxbridge interviews is to push you past the curriculum covered in schools, and to see how you respond in a one-on-one academic discussion. So a student might have memorised the entire text book and mark scheme, might be getting full marks in all exams - but still not pass an Oxbridge interview, if they are not good at analysing academic material on the spot and expressing their thought process aloud. So preparing a student for A-levels is a very different business to preparing them for Oxbridge interviews.