I think it's unlikely that many of our cyclists who have won most of our medals went to private school.
Traditionally sport has been neglected by UK governments since time began. Hence the only people who cost afford to participate as adults where those drawn from the better off.
Secondly sport has often been socially perceived as a pastime rather than a competitive activity. Don't forget tennis was the speed dating of the pre war era!
Compare this with the fantastic state sponsored sports system in France, it's a wonder we won anything. Closer examination and you'll see our more successful sports traditionally have been high entry cost sports usually involving a boat, gun or horse - not particulary inclusive is it?
However there has been some outstanding successes over the last decade with the direct assistance of funding from Sport UK, Sport England etc. Cycling is the obvious example going from nowhere to the No.1 country in the World. This direct, targeted funding has created a network of facilities and coaches able to support our young athletes in many sports.
Obviously supportive parent(s) are incredibly important to this process but that is the case irrespective of their socio-economic profile. To suggest that Team GB's amazing success is solely due to the efforts and sacrifices of one particular group is simply fatuous.