Shooting the concept of 'privileged' perhaps could be used in the wider sense then. Of course Trades Unions and other organisations of representation and power are closed and suspicious of newcomers, especially TU negotiators! It may also be a 'who you know' culture but that's just another political game that you have to play. Ed Milliband's education didn't make him PM but his Dad's intellectual dinner parties definitely did. Another kind of privilege.
Political power and power in the professions is far more based on where you have come from, whether you are safe, the right kind of person. The more power there is the fewer risks they want to take. And that's the nub, diversity means dealing with people we don't understand.
I think what John Major objects to is the lack of social movement. Children that go through school without advantage, whether it is pushy parents or pushy schools, do not move ahead.
Changing school funding would help but we would break this nonsense up completely if we enhanced the status of vocational skills. In Germany it is quite common for one child in the same family to go to the tech school and the other to the academic school. Neither child is more privileged than the other. Parents are proud of both children equally, both children earn roughly the same money because the educated professions don't cream off the cash from the vocational and skilled workers. A focus on equality in the labour market would really help I think.