Xenia I think the choice of peer group is much more nuanced than 'private good, state bad' as I imagine do you. I expect, with that in mind, you chose the particular school carefully.
I know my experience growing up in a small, northern, university town isn't typical of many places but, as mentioned above, my friends at the local comp were children of professional graduates with high expectations, confidence and awareness of the university system.
Peer group and population are not the same thing, there were also lots of children with different backgrounds but enough of us to form a critical mass and enable us to get on with fulfilling expectation unimpeded, also creating a culture of expectation that others could join in with.
The local private girls' day school was not especially academically impressive, in fact my comp's A-level results were often better but I think girls were sent there precisely to mix with a perceived socially desirable peer group and avoid a perceived less desirable one. Many of the pupils were children of local business people who themselves probably hadn't been to university. In this instance these children probably needed the greater support with university applications offered by their school as they didn't have that academic capital at home, as we did. I was happier with my academic peer group.
One other thing I'd mention on the general topic is I have met people who were hot-housed through A-levels and propelled into good universities by private schools, only to fall at the first hurdle of independent study and struggle badly or drop out as they'd essentially over-achieved at school and weren't up to the demands of their course. It may have looked good for their school to have got them in but I don't think it was in their best interests.