The effort involved in means testing, including verification, would be extremely taxing for an organization the size of YC and doubly so in relation to the size of the money involved. I have to suspect they never seriously considered it because it'd be so impractical, and there would be inevitable abuse (each layer of complexity makes such things worse, for YC simplicity has always been integral).
For my grant I was asked to sign a simple declaration of means. There was no verification. It was more to do with the aspiration of the Trust rather than an ironclad requirement (I was funded age 15: the minimum was supposed to be 16).
There would be no additional burden on YC to say 'we want to encourage more people from underprivileged backgrounds to start a company'.
At this level of funding YC's impact is going to be much greater for that demographic than those whose parents can afford to seed their startup just as they'd buy them their first car.
