Haven't read all but comments from my state sector (comp) experience are:
Looking at my infant and junior school counterparts, about 10 to 15% of whom went private at some stage, everyone who clearly had potential to go to Oxbridge, other top universities and med schools did so regardless of sector.
There were a few exceptionally able people who went to the state school and chose the better or more interesting course at a good university, rather than applying to Oxbridge. I wonder if a private school, more concerned with their own marketing, would have pushed them towards Oxbridge for the sake of their ratings.
The people whose achievements surprised me, positively, were the middle-ability children who went private and did better than expected. Similar people in state schools did ok, or drifted in with the wrong crowd and did badly, they weren't supported or saved by the staff in the same way.
Almost all my friends at comp were children of university lecturers, teachers and doctors. The first two professions were not well paid enough to go private and lots of left-leaning parents wouldn't have done it if they did have the money. So parental expectation and confidence were huge. In common with many private schools I wonder how hard our teachers had to try, there is no great challenge in making a silk purse out of a silk purse.
I think we should not underestimate the power of the peer group. You think you chose a school for its ethos. I think the attitude of your child's friends is at least as important.
Later in life I've met a number of Oxbridge graduates whose parents were not university educated, most of whom went to state school. The parents valued and encouraged education. The school encouraged the particular applications.