I don’t think superiority is explicitly taught in the way some think it is, although that may depend on school/family. You’re not taught that you’re better than others, but what is instilled is self belief. Of course you’re going to go on and be successful/rise to the top in your field, because why wouldn’t you?
as far as manners and charm go - yes, you are expected to be able to conduct yourself socially in accordance with environment. Confidence though, not arrogance. Had a few teachers that could cut you down with a sentence if you displayed the latter.
Whether people go on to be ‘charming’ or not is another matter. Some will only employ it when they deem it beneficial to do so, whilst being dismissive and rude to those they perceive to have no benefit to them. I don’t think that’s a character trait exclusive to private schools, however, and I don’t believe it’s encouraged either. At least it wasn’t in my experience.
Some kids will go to private school but aren’t too worried about academic achievement. They’ll get the grades they need, but won’t grind for the highest distinctions, because they don’t need to. Academic achievement is just a formality. Their path forward is already set, and often results in them leaving university and getting a position in the family business/the business of a ‘close family friend’. I don’t think private schools are responsible for this, however, rather it’s down to the family and the circles they inhabit.