Mominatrix and cake, London school runs can be fun!
I agree that each nationality can be pushy in its own way. The French can do it so elegantly, and I also like the way the Russians do it so honestly.
I am not sure I spotted anyone sneering at kids in the local comp. My feel is that London is tough for most people, so there is no room for smugness. A fair proportion have only gone private because local state alternatives are challenged. Indeed I reckon, based on conversations about tutors, a good proportion of my daughters secondary had narrowly missed Grammar, and this will apply to some of those who enter Preps at Yr7. If we had been in another part of the country private would not have been on our agenda, and even in London it was because we did not want to go down the god, renting or tutoring (often all three) routes that our friends who opted for state invariably had to do.
The great thing in London is that no one sees your car (ours is 13 years old) and by Yr 7 it is not cool to be taken to school by parents. No one really sees your house either (redecoration was something that had to wait as we prioritised fees), plus a small flat/house does not tell you much. For all you know there is also a country estate or a chateau. Most mums at DCs prep seemed to work, many because the money was needed. (Not true of all preps, but then each school seems to have its own culture.)
I have heard that outside London the school run can be different. That people rock up in huge tractor like cars and talk about the golf club, kitchen islands and spa days. I would not have fitted in, but then mine would have been at the "bog-standard" comp so it would not have been a problem.
In my experience Beneden is seen as a lovely school. Not all parents have Wycombe Abbey type daughters, and I suspect few parents see this as a bad thing. Similarly visiting the dementia suffering mum normally promotes envy, especially for expats who worried about parents in another country/continent who are struggling. While ski trips don't get mentioned for fear that you get top trumped by the parent with a Swiss Chalet.