I certainly can't disagree with any of that.
Perhaps I was just fortunate. My mates are some of the most down-to-earth people you'll ever meet, and we were (without meaning to sound utterly ridiculous) the 'cool' ones. Some of us come from wealthy backgrounds, others of us have parents who have decided to live minimally because they believe that it's worth the sacrifice to get private education for their children. None of us are snobs, and in fairness I didn't see too much of it amongst the boys at all. The girls were a slightly different kettle of fish though, in fairness, so maybe it just varies.
Having said that, I have met the detestable types at uni, but all I can say is that their schools must have been different to mine, and I think that's what it comes down to. Would I sent my child to Eton? No, but there are plenty of good fee-paying schools that I would send my child to if I felt it would offer them a better chance in life. Private education does not necessarily equal snobbery.