Depends on the school, and on the child. Impossible to generalise about private schools, just as it is about state schools.
However - allow me to generalise a little, based on my knowledge of quite a lot of private schools. They will often be set in larger grounds than state schools, which can often mean more sports facilities. It can also mean dedicated music blocks, art blocks, theatre/drama studio, space to keep chickens/guinea-pigs/rabbits, gardening etc etc ... There's usually something for everyone. A chat with the Head, and a look around the school, should give you a pretty good idea of what the extra-curricular focus is, and of how big a part sport plays in the school's daily life.
My DH is a PE teacher in a private school, and I can reassure you that PE these days is VERY different to how it was 30 years ago. It is much more humane, there is a huge range of sports on offer, and teachers really do try to encourage all students to find something that they can enjoy. So even for a non-sporty child, the sports available at some private schools really can lead to increased confidence in other areas of their lives. Gone are the days when a bearded man in skintight nylon trackie bottoms bellowed at you through a megaphone while you struggled round a 400m track 10 times in blizzard conditions
These days kids can play table tennis, do all manner of less strenuous activities such as golf, archery etc, and even go out and do kayaking, sailing, riding, surfing etc ... when they are old enough they can do weight training, general keep-fit activities, Zumba etc ... and cricket and rounders are lovely for the non-sporty because most of the time is spent standing around chatting whilst waiting for your own, very brief, turn!