When I was running the city teams in Sheffield (back in the late 1980s) hardly any players nominated by the local independent schools were ever good enough for the city (or county teams). Nearly all of the best players attended local comprehensive schools. They included our Under-15 captain, Michael Vaughan, who was a pupil at Silverdale (Comprehensive) School.
However it's fair to say that the best players usually came from families where there was considerable involvement with local club sides, who got youngsters involved with coaching, and playing competitively, from a very early age.