I think that, as other people have said, if you have a long family history of Oxbridge, your parents are likely to be wealthy and likely to have provided you with an exemplary education in an academically-competitive environment. If A-levels were a test of raw intellect then this wouldn't make a difference, but we all know they're not - it's about how well you're trained in the technique. Luckily, there's the interview process, which helps them to spot people and, I think, guards quite well against people being "trained".
People like me - single-parent low-income family, completely unremarkable mediocre comprehensive school with no Oxbridge tradition, Cambridge offer - are proof that the system, if not quite perfect, is definitely changing for the better.