Most college students are not going to get admitted to Stanford or any similarly generous private university (which is relatively few of the total number of universities available). And those from lower income families have a bigger hill to climb to earn the merit that would make them realistic candidates for admission (these universities typically get about half full-pay (top 5% family income), with Pell grant (bottom half) being significantly less than half of those who do get FA).
In terms of state schools, some states have relatively good in-state FA, but others do not, so students from lower income families struggle to afford their in-state publics in those states (Pennsylvania is the usual example, but is not the only one).
Most college students are money-limited in their choice of college (or if they go to college at all), so most commute to colleges local to where they lived before college. To have more of a choice, most students would need either parents with money, or top-end high school academic credentials.
https://www.census.gov/quickfacts/fact/table/losangelescitycalifornia,newyorkcitynewyork/PST045219 says that median household income in Los Angeles is $58,385, and in New York it is $60,762.
In any case, upper middle income families can easily make themselves lower income by taking lower paying jobs and giving their assets away to charity (with enough lead time due to the prior-prior-tax-year used on FAFSA and CSS Profile). It is usually a lot harder for lower income families to become upper middle income, despite most wanting and trying to do so.