No doubt that poor students benefit from the well resourced private institutions if they are lucky enough to attend one. However, the vast majority of the students in this demographic, if they attend college at all, usually attend a CC or local 4 -year commuter college.
I teach at a 4-yr commuter college in the Nj metro area, and although our graduation rate may not be as high as a highly selective college, we still manage to graduate a fair number of students . My university is an eclectic mix of students of various ethnicities- but they are all primarily of low SES background. Many are happy to get a job that’s better than what their parent(s) have. I think this is what @MWolf was pointing out. BTW, I don’t think that even our accounting majors would know what the Big 4 accounting companies are! They’re just happy to get a job at some local business, or be promoted at the store where they were working part time while attending school.
This is not the college universe of many of the parents here on CC, but it’s the reality for many others. And the persistent narrative that the Ivies and their ilk are the best at bringing up those from the bottom tier is simply counter productive as it serves to dilute support for public higher education.