I don't think you have any idea of what the makeup of private schools is like in today's world. You are stuck somewhere in the 1980s.
The top private schools in London today are at least 50% ethnically non-white. About half of all the kids are of Asian (Asian and South East Asian origins). Of the white kids, most are not 100% British (including ourselves). At least one of the parents is not British (quite a lot of East European), and sometimes both.
And the main thing they have in common is that they are grafters . These are kids who have been put under pressure and pushed from the start - I am not saying that is a good thing, but it is an undeniable fact! In my DC's class, one child plays in the Academy of a premier league football club, another is the top tennis player in England for his age, another one is some sort of snooker champion. At least 50% play an instrument at Grade 4 + (they are 11 years old). Many go to language schools on Saturday morning (Arabic, Russian, Flemish, French) as the parents want them to speak their home language properly, not just English. These achievements do not just fall into your lap, however rich your parents are, and however many tutors you have. It takes hours of work or practice, and not all children are able to manage it, whatever their background.
The kids are confident, but humble - they are well aware of the desire everyone around them has for them to perform.... and that desire to succeed has become ingrained in them. If anything, they often doubt themselves, as they are so used to having to compete with other high performing children, every single day (even in the top schools, the kids are put in sets, and live in fear of dropping down one), and can never let performance drop - which is sad to see.
Whilst it is undeniably terrible that so many kids have been deprived opportunity for so long, two wrongs don't make a right, and it is sad to see such prejudice against kids whose parents have often battled up the social ladder themselves (interestingly, a lot of the parents at my DC's private school are themselves the children of teachers, who did well academically and then professionally) - and who in the majority of cases are making financial and lifestyle sacrifices (very few SAHMs) to send their kids to the best schools they can afford. These kids may be priviledged - but it doesn't make them entitled, and it doesn't make them stupid, either.
Would be good to move past the cliches and update the knowledge to have a more informed debate.