Whatever else you can say about Abbott, she's not exactly a toff from a privileged background. Besides, she read history, not mathematics.
Back when I applied in the 80s, it was definitely easier for a comprehensive student to get into Cambridge rather than Oxford. You didn't have to study extra material for the entrance exam - you just took the STEP based on the same syllabus as your A levels - and you'd almost certainly get an offer if you were studying a STEM subject, even if it was AAAA11. Once there, it was pretty much down to how talented and prepared to work you were; the faculty didn't show any favoritism to those from public schools, so far as I could tell.
Nowadays, they've both opened up a bit more, even if there's still more to do. You can just apply via UCAS without having to submit a separate form. Oxford now also uses entrance tests that don't require extra study. Cambridge still seems to have the better reputation for access though; with more former comprehensive students having studied there they've at least got a slightly more diverse alumni network, I suppose. If anything, the biggest barrier is that a lot of talented students tend to count themselves out; there are a lot less applications per place for Cambridge and Oxford than many other universities because they don't even put one in.
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- Sergiy Kyslytsya