My impression is that people who currently send their kids private would see any of their children doing such jobs as failures and would very very strongly discourage them from pursuing them. Ultimately, if the expectation is your aim is to make money then kids won't go into nursing or teaching.
I wouldn't see those occupations as failures (not least because they're probably contributing far more to society than my job). I think it's more kids perhaps wanting the same lifestyle as their parents, who are usually reasonably well-off to be able to afford the fees.
My son's dream job would be a personal fitness trainer but he's pragmatic and realises that, if he chooses that route, he's going to have a more financially constrained lifestyle. Nothing wrong with that but that's not what he'd choose.
Our friendship group mostly work in financial or professional services, so he sees that as the route to being able to afford a flat in London, nice holidays or whatever. We'd support whatever career our kids choose so it's coming from them, not us. I also don't think that's a private school thing, it's probably common in higher socio-economic areas.
Above all, my kids have loved being at their school because the sports provision is brilliant. That's made it a worthwhile investment for us, together with the very good academic results the school achieves (as do some of our local state schools).