I got my doctorate from and then was a lecturer at Oxford pre-DS.
Can't speak for the other place, although I have friends and a grandmother who went there.
Oxford students are bright. Some truly brilliant, some are bright as the consequence of really hard work. They all work hard while they are at Oxford.
In general (with a couple of exceptions) Oxford Colleges couldn't care less who your parents are, and are actually really keen to take students from State schools.
BUT they have a couple of big problems:
It can be hard to tell the difference in interview between someone who's, frankly, not up to getting in, and someone who has amazing potential but has been at a mediocre school, without necessarily a lot of encouragement at home and who is completely intimidated by Oxford surroundings. It's a real interview skill to be able to pick those people out.
Also, there are issues with bright students from certain areas/schools never even considering applying to Oxford, because it's just not part of their world at all. Doesn't mean they're not as bright or brighter than the students who apply.
As for it being relevant or important that most of the cabinet came from Oxbridge. Pah. A lot of people who are academically gifted are pretty unworldly. That's why they call it the 'ivory tower'. Yes, they are likely to be capable of hard work and of focusing pretty intensely on tasks at hand, but their organization and social awareness can sometimes be (ahem) pretty average.
Stupid example: only one person in my lab of 7 had a TV at home. Lots had no idea what was going on outside their plants/fish/insects.