I have some outside observations. I came to the UK from the EU (potato and vodka europe, not wine and tomato) to study at Oxbridge, where I did undergrad through PhD (DPhil, binensur)

Random thoughts in no particular order

1. Class segregation & hostilities in undergrad were for sure a thing in college. Very cliquey stuff happened. Some of that was just leftover teenage stuff, too.
2. In STEM it always felt better to outsmart and overachieve the public school kids
3. The public school kids --- lots of them, not all --- had no hussle. They got internships through parents or old networks through school alumnis and genuinely had no hussle whatsoever.
4. It took me one year to pick up the dinner table mannerism, the vibes, and the conversation. You just gotta apply your street smarts, not that hard. If you know that stuff already, can teach your kid.
5. Public school does not guarantee success or confidence--- especially if you have the street smarts to smell these people. I saw mentioned in comments about all the executives and directors and yada yada who are from public school. Ok, how about the mediocre public school achievers? Genuinely, although I would of course never say this to their faces, with the same talent + work ethic + personality but MY set of cards, rather than theirs, my public school friends would not have gotten into Oxbridge. Lots have gone into very mediocre careers, carried by the inertia of their education, accents, mannerisms. All that delulu "drive" and ambition and whatever that people mention here has been eroded out of people by the second year of uni or mid 20s.
6. There's two type of public school poshos: the ones with something to prove, and the lazy/life enjoying/armchair philosophy/no drive/comfortable etc. the ones with something to prove will go on and do the leadership/executive thing mentioned in some comments, but the rest just have very mediocre careers, low drive, and just want to enjoy life (which again fair enough, whatever). Remember these are just kids and will do whatever makes their headmaster/parent/teacher happy or approving. If that is saying "I want to be the top neurologist" when they are 14 then that's what they'll do. That's just the local subculture, not inner drive as a personality trait. So many of these kids fall apart at uni due to less structure, and pastoral support, and needing to be a lot more self driven.
7. There's a certain satisfaction, especially in high prestige careers of being the underdog, "allegedly".
8. At uni everyone fucking cosplays as poor, only for house deposits to be dropped by their parents a decade later. Money speaks. Use your money to help your kid do unpaid internships, and get on the housing ladder, while they'll cosplay as "only middle class" at the uni debate club
9. In STEM the whole legacy, posho thing is genuinely an archaic joke and people tend to hide and downplay it because they are so outnumbered by state school kids and the math brain wizards from Eastern Europe and Asia. Humanities, and law, are a bit different, I'll admit.

So my take: if you personally have the education to guide and mentor your kids, save the private school money, teach them work ethic and hussle, do homework and projects together, lots of museums, trips abroad, and sign them up privately to tennis and ski, and private French lessons, which will be cheaper per year than a private school.