I suspect it's a mix. Private schools do a lot more sports - which gives kids fitness, general sporting capability, exposure to a lot of different sports - and have specialised sports teachers and a culture of taking sports to competition level. They train hard, and take teams of kids to national competions from an early age.
Some children and parents in the state sector will of course have the knowledge and drive to do it themselves, but some (who may have become top athletes with PS support) won't.
Private schools are also much more relaxed than state schools about children taking time out of school for high-level involvement in sports, music, acting etc. It's part of PS culture to support children individually to succeed in whatever their 'thing' is - not only academic - and they have the staff to still get the kids up to speed on academics in the shorter time they're in school.
But yes, it's also parents. Getting to a high level in sports takes money for training and kit and transport to competitions. It also requires enormous amounts of time and commitment from parents as well as children. There will be a correlation between parents who having the resources (financial and personal) to support that and sending kids to PS.
Controversiallly, I'd also suggest that high level sporting success often involves parents having drive and a desire to succeed even at high cost - high level sports really does exact a high personal cost - which I suggest will also correlate with them putting their kids in PS. Note: only correlation. There are of course people of all types in both sectors. But parents with that drive will more often be successful themselves, and also ambitious for their DC's success in all areas.