The state/private debate in this context is interesting. My DCs attend a small, academically selective independent, but not one which produces very obviously "public school shiny" applicants for university. By that, I mean that few of the pupils are from very wealthy families and there is little evidence of cut-glass accents, expensive hobbies and "cv-buffing" extras that would not be available to a state school pupil from a similar background.
The experience of friends who have DCs in the state system - be it our local comprehensive system, or grammar schools in areas where that system operates - is that the initiatives and benefits rolled out in the state system which are intended to level the playing field, are snapped up by pupils who don't really need them - the DCs of university-educated professionals who have chosen to use the state system, but who have the opportunity to benefit from a lot of support from home - music lessons, extra- curricular activity, advice on subject choices and careers, help with personal statements, connections for work experience.
I am told that it is far less common for a pupil who is academically very able, but from a less advantaged background, to be given the practical advice and help they need from an early stage, to ensure they don't limit their choices. I'm talking about pupils who don't have access at home to advice on the best facilitating subjects to choose, and which courses and universities will give them the best range of options. It seems that some of the initiatives, such as our local scheme to introduce bright state school pupils to Oxbridge, are dominated by the offspring of sharp-elbowed professional parents, and those for whom the experience really could be life-changing, still think it is not for the likes of them.
Anecdotal evidence from friends with DCs also going through the system, is that unless the school is in a deprived area, the fact that they attend a state school as opposed to an independent, has not afforded them any particular benefit in securing offers from universities.