My dd isn't particularly sporty and is not one of the kids to get picked for school teams etc. Personally, I think that's fine - it makes sense for the school to pick the best players, and if some kids are particularly talented at sport, why shouldn't they have a chance to shine?
That said, dd's school does have lots of opportunities for recreational sport clubs that are open to all children, and indeed, all children are encouraged to join in.
Also, dd is one of those kids who gets picked for a lot of other stuff - she is a very confident public speaker and so tends to get chosen for school plays, speeches, positions of responsibility etc. She is also very academic and so she had lots of opportunities to do G&T enrichment activities that other kids might not have a chance to do. And she is centre stage for every dance performance that she ever does. I actually think that learning how to cope with not getting picked for certain things is a positive lesson for her to learn - she may be very good at lots of things, but nobody is good at everything and that's ok.
Her school is very good at recognising kids who are good at art, music, maths, creative writing or whatever - it isn't just about sport. There are also awards to recognise other contributions, such as kindness and consideration towards others, perseverance in the face of obstacles etc.
Yes, most of the inter-school competitions are focused on sport, but certainly not all - dd has competed against other schools as part of her school choir, then there are chess competitions, creative writing competitions, art competitions, sciencey-type invention competitions etc. This is a state primary, competing mostly against other state primaries.
I would like to think that all kids have a chance to shine in at least one area. It makes me sad that people might resent dd for getting chosen for the stuff that she is good at. There is plenty of stuff that she can't do, and I am very happy to celebrate and acknowledge the achievements of other children in those areas.