If you actually bother to look, there is no defining public school pupil or parent. There are, unfortunately, people who fit the "stereotype" but they're often merely tolerated by peers. Most of them out grow it and grow into perfectly nice people. There are also people at public schools who fit plenty of other sterotypes, too - again, perfectly nice people
I'm not sure what stereotype you mean? I'm not from that world, I went to a state comp, but I have 'bothered to look' at the stats for different universities and professions, and I also have visited some public schools for work reasons, and know a lot of people who went to public schools, including a few current pupils.
If I was looking for common characteristics I'd say ex-public schoolboys tend to be very articulate with extremely good social skills, and are confident bordering on arrogant about their ability and knowledge. I've met Boris & know a couple of people who have known him through work, and I believe he is very much like this. I think the social skills and bullet-proof confidence are cultivated by public schools, but, as in any group, some will be altruistic, honest, capable, others will be incompetent, self-serving and dishonest, most probably in the middle somewhere. But great confidence is a problem when people are unaware of their own shortcomings, and it's dangerous if those people are in positions of power. I think that is maybe what the OP was getting at.
The other big problem I've seen in my industry is that the public school network throws up huge opportunities through extensive contacts, internships, people in powerful positions preferring (maybe unconsciously) to give jobs to people like themselves. In many jobs, including ones on your list, there is an over-representation of people who were privately educated. That's sad for able state school kids who can't break into those professions, but it's even sadder for the UK, economically and socially, if we aren't drawing on the full pool of available talent because of networking by people who went to a small number of schools.
I don't think it's fair to demonise the people who went to public schools as if they are training grounds for dishonesty and evil. That's obviously nonsense, but I do think a democratic country should be concerned if there are institutions which skew power and influence, sometimes allowing mediocrity to flourish.