@Tootletum well that's the interesting thing and something my DH comments on all the time (he was in state education before transferring into private for sixth form and overall is much more intelligent and accomplished than I am): he can't believe how much self confidence and authority me and my friends have both in how we view ourselves and in how we interact with the world.
This manifests itself in simple day to day interactions where I have no problem questioning or pushing back on doctor's or other people in positions of authority or at parties where I feel confident vocalizing opinions on things I probably realistically only know on a superficial level.
When we were deciding what education path to go for our children, for him this (unmerited) self confidence was what pushed him to agree with me to go private all the way through. For him it wasn't necessarily about the education, it was more about the "soft factors": what happens within a system that produces children that are utterly confident and convinced that they too deserve to be heard and respected.
This is a very long winded answer to your question but when I ended up at Oxford (at the time), I was utterly convinced I not only belonged and deserved to be there, I think I was full of the teenage hubris about my own "brilliance". I remember thinking at the time and during much of my 20s as I worked in my career, "well everyone I know gets X or does Y, so why shouldn't I be admitted to Oxbridge, or get that job offer because I'm no worse than they are"
I do think at Oxbridge a lot of the confidence came from possessing the right social capital and knowledge and less about school work. Knowing about film, music, politics, literature and being overall well traveled with the right funny anecdotes carried a long way in day to day life.
For reference my parents were an academic at a university and a solicitor. Both fairly successful in Their fields. I am grateful that they gave the resources and space for me to feel like I flourished and only now in middle age do I have a more honest perspective.
As for my kids, I do hope they end up being happy and fulfilled. I would be disingenuous to say that I don't care about their financial well being but I do. I think the way the education system is set up, it funnels potential high earners together and thus creates more chances of them meeting their life partners at uni or shortly thereafter. And so the cycle of it all begins again...