The problem is to design a system which allows talented kids to advance regardless of the success or failure of their parents.
It's difficult because a)we are not all that good at measuring talent, and b)talent florishes or dies depending on opportunity and we all know the deficiencies of public schools in poor neighborhoods.
I don't know the answer but I went to school in the '50s and '60s when a good education was easily obtained - at least in California.
As much as I dislike the idea of the state lottery (the government shouldn't be in the business of converting the people into marks), here in GA, the Hope Scholarship seems to have done that reasonably effectively. Maintain a "B" average, and the gov't pays your tuition. Lots of kids who otherwise wouldn't have been in college are there. SAT scores for entering freshmen at UGA have increased markedly, is one result, so there's the "allowing talented kids to advance" part you spoke of.