In the majority they don't, no. A lot of it is down to teachers of yester year being prepared to give up their evenings to take the school sides to play other schools and the availability of playing fields, but both are in short supply now for different reasons. It's also got more expensive so schools seem to have given up on it and the ECB don't seem to see grass roots development in traditionally working class areas as a priority.
I'd disagree that the current side don't have hunger, many privatly educated students represent us very well in the Olympics and in Rugby Union, but it's too small a talent pool.
It's the ECB who should be blamed.