Had a debate in RL with a few mums about this a couple of weeks ago - mix of private and state educated, but all our children are private. Private educated mums didn't even really 'get' what the state mums were saying. Really had no idea of extent to which the state sisters had had to 'fake it till we made it'.
All the mums in the group were, I guess, alpha mums (v successful in our very different fields), but only one had been a typical alpha girl at school. Yet we'd all done great at school, gone to the best unis, and generally trod very similar paths, wih really full lives outside of work. But the big difference was that the state ed mums had all felt they had either held themselves back at certain times, or had forced themselves to go against their natural inclination in order to 'go for it' whereas the private ed mums couldn't get their heads round this fear of failure (they were quite shocked when all the state mums admitted to it).
What was interesting was that the debate came about as part of a discussion about indie vs grammar. The state mums all firmly believed that £35k PA was a price worth paying for the indie option, whereas the private mums were very confident that state would be fine if the schools was good. Things like peer pressure to not appear clever, potential lack of constant affirmation, potential drop in competitiveness, potentially different parent-student relationship, and possibly less discussion-led teaching had not occurred to the private mums, but were all cited as perceived differences between state and private by the state mums (wrongly or rightly) and were considered to be the building blocks that lead to the confidence thing, as was the concept that in the good private schools kids aren't allowed to fail academically, and it would be considered that the school had failed, whereas having kids not reach their potential was possibly an accepted part of grammar life.
Debate ended, by the way, with the state mums all even more concrete that they definitely wanted their kids to end up like the private mums in the group, with totally unshakeable confidence, and that it was a price worth paying.