You can't "raise yourself" by your own bootstraps, and you cant do it when your hands are tied. Given how deep the problem runs, it is amazing that many people in poverty actually get out at all. The problem is ingrained in the system that feeds it. As said, flight is just one example of passive programs that create a chain-reaction leading to an implosion of the quality of education for some demographics.
Until the problem is confronted, addressed and exchanged for a solution, it will always be there. And, the thing is that it benefits a large(r) group of people, so the "problem" itself usually goes unnoticed.
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Your last sentence is, unfortunately, a "proverb" of the problem. The American economic system isn't fair for anyone, so what one can afford is based on one's socioeconomic background and current placement. This, again, is a fundamental part of the problem.
The problem is 1) ignorance of the problem, and 2) a cowardice to address and tackle the problem.
The problem is very clear - the connotation of ignorance being that people make a choice to ignore something when they have the means and opportunity to learn/find out about it.