This thing about snobbery - you know, although there are some snobs in all walks of life, I think that this sense of feeling like someone else is looking down on you all the time, is often imagined and some people are more prone to feeling like this than others - which explains why even in the same school people can feel they have had very different experiences.
Some people are very aware (and bothered) by having less money or a different background to others - they are hyper sensitive to things others say or do - and often read judgement or snobby attitudes into the behaviour of others, where it isn't often there. Other people have exactly the same amounts of money and background, but don't get this sense of being judged or looked down at for their car or their house or their background, perhaps because they simply aren't that bothered about those things and don't place those things very highly themselves, so aren't looking for people to be critical of them.
All I can say about this, is that it is worth being aware of yourself and which kind of person you are OP, in terms of whether you are likely to feel people are snobby towards you or not. This will depend on how confident you feel in yourself and whether being with people who perhaps have more money than you, or a different background will make you feel judged or looked down on. If you think you might be in the latter camp and particularly if your child has picked up on these kind of feelings and taken them in themselves, then being in an Independnet school might be difficult. If however, you have a sense that you are as good as anyone and that your and their value isn't related to how much money you have or your backgrounds and you approach and expect to be approached as an individual in a world where most people are nice and decent and easy to get on with....you will be absolutely fine. Anyone entering a new schoolsas a parent at any type of school does better in my view, when they are friendly and open and simply expect others to behave in the same but if there is an expectation that other people might not treat you like that, you start to see or perhaps imagine people excluding or being unfriendly or snobby or whatever.
Just an observation, in light of many of the comments on this thread and being the poorer person or person from a different background.