The networking that gets you into politics and unadvertised jobs is done among a small elite group that is a minority even at Eton etc. ie it's not that Eton creates the elite; it just happens to be the school the elite go to.
I went to private schools but my parents didn't pay for most of them. I couldn't pay for similar schools for my kids - well, I could have had one less child, but I didn't, and it would be a long commute for them, and a lot of anxiety - I have health problems so knowing I can give up work again at any time is a huge load off my mind.
On the whole, the main difference is bad news stays out of the papers, followed by many more on-site activities. Some friends with similar age kids go private (live in small towns) and are very envious of the number of school trips my state-primary kids get (it's London, but also places know our school will leap at opportunities so sometimes a museum will call and say 'we've had a cancellation - fancy bringing a class to an exhibition on X in three days time').
My boys have ASD so struggle with many aspects of school, especially sport, art and music - so could say it's handy they aren't forced to do much of it. At my private schools it was always a worry for many kids that if they didn't do adequately academically, they'd be kicked out - not a concern in state.
Local teens have said they don't see the point in grammar schools because the local comps are so large, once you're in sets, it's like having a grammar and sec mod on one site, only with classes and pupils who get to do both.
I figure as long as there is a cohort of bright kids who want to achieve, and a larger proportion from families who take education seriously, state plus parents supporting interests will be OK. Which so far we have - loads of my kids' peers come from families without much education themselves, but will damn well make sure their kids get one (mainly immigrants). If I was in a small town, I might have decided differently.