We've had kids at private and state. Our privately educated DC has turned out well but so have the other two. Private one has slightly more social polish, but that could easily be a personality thing.
Re sport, we found ours got much higher quality coaching from town clubs than from private school. Might have been a different situation if they were A team material, but as B team, they got any old history/science teacher that vaguely knew the rules, whereas town teams, if you do your research and choose well, are much better. On the flip side, as a B team standard he wouldn't have found it easy to stay in a good quality town team throughout teen years, whereas the private school at least forced him to find other forms of exercise, and luckily one of those has stuck with him into adulthood.
Teachers were a mixed bag at both schools. We had to pay for a maths tutor for privately educated DC because of a run of bad teachers and bad luck with staffing. Bright DC in state school had slightly more low-level disruption in classes generally though, and less bright DC in state school, in lower sets, had a lot more disruption.
Add to that that uni offers will often reflect the different educational backgrounds of applicants, so even if private does notch you up a grade, that doesn't necessarily do you any favours.
For me, the private vs state debate comes down to parental availability. If both parents are working all hours and can't take kids to extra-curricular clubs, private is a good option. If you can do the taxiing to activities yourself, you'll have a wider range of options than any school can offer, so private doesn't offer enough advantages to be worth the money.
Private also has the potential disadvantage of you coming out of school a bit more sheltered (and possibly arrogant) than the average kid. The old boy network, apart from in a very few schools like Eton, is not really a thing. Family's social connections are much more significant.