My parents went to museums with us. We went to summer camps that provided educational activities (fun physics experiments and the like). We got home schooling for learning instruments and our parents were able to help with homework and satisfy our scientific curiosity. And of course we learned the soft skills of how to move in the upper middle class environments, how to approach job interviews and so on.

These aspects help tremendously in striving for a higher education and well paying job. It is also quite interesting to see in my families history, where my parents were the first to get an academic degree, but the grandparents of my parents were already skilled craftsmen, one with 4 master titles in different metal working crafts.

There is only so much the government can do and i'd say that two to three generations are probably the bottom line for how fast it can go on a societal level. In the other countries like Germany there is systemic class discrimination to keep the lower classes low.