Why are you thinking in terms of class? Class-based conflict was the center of original Marxism. Then the 20th century Marxists, realizing that it wasn't working because capitalism was actually good for people, moved on to identity politics (neo-Marxism), with which they've had far greater success in disrupting and tearing down our society. And, now that there is finally some institutional pushback against neo-Marxism, you want to go back to classic Marxism?
I think your worldview needs to be updated to not be centered around group-based dynamics and politics. Individual meritocracy is the least corrupt system we've come up with to determine access to opportunity. How about we gate access to Harvard based on things like ability, and focus instead on making more high quality schools? This would increase the supply of quality education, which would actually make the world a more equitable place! It's amazing how many "equity"-based policies exist that actually make the world a less fair, less equitable place!
Part of the reason the above is so difficult is specifically because the institutional elite of our day want a class-based society. They want to be part of an emerging new nobility class, and they like having certain filters and gates like ivy league schools. This is what Marxism has always devolved into, really. Harvard loses it's power when we stop treating it as so special, and that goes hand in hand with fighting against those in power today who want to build a an ever-wider moat around their power.