Someone sufficiently poor probably doesn't have the funds to start a startup period.
You might strike lucky and get funding from YC but how likely is that to cover everything? Including getting to the stage of securing the funding (like travel expenses, time to build a prototype etc). Also amongst poorer families there is a greater expectation that children will contribute back to the family pot at an earlier age.
I agree that it sucks to be poor, but I don't know how much startup funding can fix that.
Also since we are talking about aspirations you have to consider the "all or nothing" attitude that's so common these days, and I see a lot of people particularly in the lower classes who are far more motivated about the tiniest possibility of a quick jackpot than a slow but steady improvement in their lives.
But then again that way of thinking might have more to do with the way the middle class (the "steady option") is disappearing than with a social fad.