You need to constrain the independent school sector in some way, whether that's revoking the charitable status or something more radical. Trying to cross-pollinate pupils from both sides of the tracks and education institutions so that you're getting a fair spread of diversity rather than having these middle-class and working-class ghettos. Collective bargaining workers coming together is a primary driver of social equality. Britain could break away from its relationship with hereditary privilege by just builders in the House of Lords.