Schools. Get rid of the private ones, or at least make it so they have to operate to the same budget per child as the surrounding area (this way it's actual competition rather than just "An alternative for rich people", and a school wishing to increase its budget per child would have to help out the surrounding area).
THIS. Put schools on equal footing. Make sure all the schools get allocated the same out ammount of resources and quality teachers.
Collage for everyone.
You DONT take away money from people because "they have to much", you gauge how much they have and tax them in relation with the rest of the populace. Letting the goverment do crazy shit like take away hard-earned fortunes from families so Richy-McRicherson can be on the same level as UristMccMigrant.
You really can't do anything not 1984isn past that, People are going to be lazy no matter what.
Even if you unjustly pillaged fortunes for the sake of stamping out that darn "privilege" or whateverthefuck tumblr crusaders are calling it. There's nothing stopping RichyMcRicherson or UristMcMigrant from hoping online and learning by themselves to get ahead. The government would need to make sure no-one has access to information unless its spoonfed to them, and no deviations of said ideas were allowed to exist. It would be a fucking nightmare.
Even in that distopian scenario, social stratification would occur.
Some people will just be better off/have an easier time than others