Interesting phenomenon. I'm not sure it is just the private schools, but I think more and more future success in sport is correlated to family income and ability to support the child in that sport.
Take for example swimming - it costs a lot to do this regularly. To swim for a club, you could be looking at well over $1000 per year, maybe even $2K. Every now and again, you might get a kid that just jumps in the water and breezes past everyone else but that is the exception, not the rule. The reality is that most kids need to be funded for a number of years to get better. If you are comparing a 17 year old kid that has had $10,000 "invested" in his / her swimming over a number of years with regular training, against a kid that might swim for a school or at home a few times a year, chances are the kid with extensive practice will be better.
Same with team sports. Holiday programs are a few hundred dollars, good clubs demand higher fees etc etc and it becomes a self supporting prophecy.
It is the exception to find a Ronaldo hanging out in the favelas of Brazil, not the rule