Yes, I suppose you're right but still it doesn't seem to happen that often with football. Probably because of the types of sports that are focused on at these schools. Plenty of posh rugby union and cricket players out there of course, and looking at this breakdown of medallists from the Olympics, it is the sports that require expensive facilities and equipment where the more well off do better.
Overall 35% of British medallists were privately educated at the last Olympics compared to just 7% of the population attending them. Football is still seen as a working class sport.