As with anything else, those people who feel the need to boast about their wealth/ education/ dcs or anything else are often the most insecure. Like combustible, I've never come across boasting/ assumption that they are the envy of everyone at my dcs' school.
Also, I agree there's an awful lot of stereotyping here. In any school you will meet parents who are obnoxious for one reason or another and that doesn't mean that all the parents are.
It's worth pointing out the diversity of private schools and fees. My dcs' school is in north of England where fees are smaller and mortgages are too. There are many parents who you wouldn't associate with private schools and probably wouldn't find it possible to afford private schools in London for example. Dp and I are teachers in state sector (lots of other MNetters like this like Martianbishop for example), I know a student whose parents pay the fees, I know a couple who work more than 1 job (including working in a nursery) to afford the fees, a driving instructor, an air hostess etc.
As for the OP, I think I understand this. I chose private education only because we live in a very deprived area where schools are dire. I didn't want to be hypocritical and opt for a faith school or move to a leafier suburb. With the dcs now having been going to the school for a few years I feel a little envious that I never had the kind of education that they have.
Like others have mentioned the school's facilities are no better than state schools and may be worse in some areas (the buildings are old and some are listed so difficult to adapt, school doesn't get extra money for ICT like state schools would so you don't see Interactive whiteboards in every classroom, there is a pool and sports halls but that's shared between all the schools- kindergarten, girls and boys prep and grammar schools). What the dcs get that you don't find to the same extent in most state schools (I speak from my experience as student and teacher) is small class sizes, individual attention, high, high expectations, very little disruptive behaviour, specialist tuition (so the dcs get specialist swimming teaching, music, ICT, PE from age 3), a real sense of community and continuity since the family of schools provides from 3-18 and teachers really get to know each child.
However, there are down sides. Huge pressure on children (having said this most thrive on this), nightly homework, a huge commitment expected from parents in terms of organisation and time devoted to homework, getting correct uniform and kit etc, a lot of competition and finding it very, very difficult to excel in any area because there are lots of kids who will go on to get 5 As at A Level/ compete for region and nation in sport etc etc.
If I'd hd the option of a non-faith, local school I would have chosen it. Now I know more about the advantages of private school I may still have chosen it but I feel that my dcs are very privileged.