Education, wealth, social capital, nurture... All of them are morally arbitrary, yet set to give some children a head start over others and bring about tangible inequalities of outcome between classes. Now, some of these things are impossible to combat; others would impose too high an economic, political, and social cost; but certainly some degree of governmental meddlesomeness in the sustainment of wealth and power by the same social strata can be feasible.
As the OP suggested, the amount and quality of education rank people in the queue for the better-rewarded jobs. There’s a vast room for manoeuvre in this field, from banning private schools altogether to boosting public spending on the state sector, passing through the introduction of “meritorious admissions procedures” -as it were- to the best-funded and most prestigious state schools…
Overpaying workers would only bring about unnecessary inefficiencies and harm the very social group it pretends to ail. One must tackle the root of the problem, not its symptoms.