I don't know about your anecdote, but I know that my experience was completely different. The interviews were purely academic, and I mean purely. Of course, I can only speak for Somerville and Christchurch, and for Physics, but I don't feel my state-school background harmed my application. I can't speak for any other colleges, for Cambridge, or for other subjects, however.
Also, you say that the exams ought to be a test of aptitude. Aptitude is utterly relative - you have an aptitude for a course. A levels, GCSEs, et al, are supposed to be a measure of aptitude for a subject. For example, if someone gets an A in subject X, they are apt for that subject. However, A levels can be somewhat of a memory test, so Oxford and Cambridge have to look for additional information as well. They're not designed to assess a student's aptitude for subject X at university Y - whilst they are a good enough indicator for most universities, Oxford and Cambridge have to differentiate between many very strong candidates, so need additional source material. Oxford have their own aptitude tests, which do test aptitude for their particular courses (in some subjects, at least), and interviews do the rest. Naturally, it's not perfect, but they (Oxford and Cambridge) seem to do alright, as their academic records show.
A-levels simply can't do the weeding required. If Oxford and Cambridge gave places to everyone who met their entry requirements, they'd run out of space. They simply have to do the weeding themselves, because there are too many AAA candidates out there.
It's a shame that A levels and GCSEs aren't the best predictors of higher-level academic success, but that's just the way it is. For that reason, certain institutions (such as Oxford and Cambridge) deem it necessary to interview candidates, and set their own tests to sort the people who they believe will do well from the people who they don't believe would do so well.
It is my personal belief that the only reason wealthier private school students might have an advantage is that their schools will give them better preparation, both in the long and short term, for applying to Oxford and Cambridge. They might be more thoroughly educated, so that the interview questions won't be such bolts from the blue to them as to other candidates, or taught how to appear cleverer than they are. Either way, I don't think Oxford and Cambridge care about financial background. The discrimination you speak of, whereby they might favour the "rich kids" out of some sort of empathy is plausible, but I really think it's "academic potential" they're looking for. They want someone who will do well on the course, make them look good, and perpetuate their strong academic reputations. Like I say, that's just my take on things - I could well be wrong.