One of my uni lecturers spoke about this and it was so interesting.
He spoke about his experiences in state schools and why he left them. It wasn't because of the students, but because he could not stand the teachers who made their students feel like they would amount to nothing, like there was no ambition.
In contrast, he told us about his experiences in independent schools and how the teachers try to push their students to their full potential, they're much more ambitious and are spoon fed all these exclusive materials to help them succeed, etc.
Like in state schools teachers are inclined to get their whole class to a grade C and are therefore unable to spend extra time on more capable individuals. The student generally has to do it them self or arrange external tutoring. Independant schools can afford to look at students as individuals as their parents are paying for it.
Anyway to answer your questions from his viewpoint, state school students do outperform them at uni because in order to get there, they had to push themselves, work more independently with less resources and so they are generally better adapted to uni.
Whereas private kids don't have that extra support their parents paid for, they don't have everything prepared for them and are no longer spoon fed everything. Essentially, they are not as prepared to take their education into their own hands the way uni needs you to do so.
He said much more but it was a few months ago now and I don't remember. I had this discussion with a couple of friends who attended St. Edmunds college and they agreed with it from their own experiences.