My DH was a private boarder and played rugby (and cricket) there, alongside lots of other sports, some of which I'd never heard of until I met him! He'd wanted to send our children to private school, predominantly for the sporting opportunities, but we cannot afford it. He is very much of the view that sport is important, and that team sports in particular are essential. It's his view, ultimately, that the quality of sports opportunities at private schools is far higher.
DH played rugby at county level and had national trials, all of which were supported by the school. His parents had barely any involvement in his sports career, it was all organised through school! Once he went to uni he played semi professionally and got himself to matches every week, again, with minimal help from his parents (although they supported him financially).
Our high school does well with sports, but I think as others have said, it's the breadth and sheer number of teams available at private schools that help to provide that strong foundation.
DS is still at state primary and whilst the school does offer sports, its not a huge amount. DS does have a tendency to be a bit of a couch potato, so we make sure he attends lots of after school clubs.
Our DD's state high school does offer sport and some teams are very successful, but as other have said, the opportunities are fewer than those seemingly offered in the private schools nearby.
Our DD is very academic and likes sport, but not to the degree DH did. She plays sport outside of school, but also has music lessons and attends guides. And as a consequence, every night of the week, we are ferrying both children here, there and everywhere, to various clubs.
Interestingly, I was never interested in sport (still have limited interest now TBH) and was much more involved in the music and drama side of things. DH takes limited interest in that, yet I think that is equally important. Really I think the most important thing is that children have hobbies, and that through those hobbies they widen their circle of friends, and experiences.
All this enrichment should be provided by schools, but lots of state schools can barely provide materials for actual lessons any more. We are lucky that we can afford to pay for our children to participate in these activities, but I know that's not the case for many people.