So our friends at USNWR computed percentage Pell Grant recipients per institution for the most recently available data. I think this means one year ago.
https://www.usnews.com/best-colleges/rankings/national-universities/economic-diversity
U Texas/San Antonio 47%
UC - Irvine 43%
UC - Davis 39%
UCLA 36%
UC/Berkeley 30%
U Texas/Austin 25%
U Oregon 25%
U Georgia 24%
Emory 21%
Columbia 17%
Ga Tech 17%
MIT 17%
Cornell 16%
Harvard 16%
Northwestern 16%
Princeton 16%
Brown 15%
Dartmouth 15%
Stanford 15%
U Mich 15%
U Penn 15%
Yale 15%
Rice 14%
Vanderbilt 14%
Duke 13%
UVA 13%
CalTech 12%
Carnegie 12%
Georgetown 12%
Hopkins 12%
U Chicago 11%
Notre Dame 10%
Wash U 10%
As you can see, it’s hard to maintain a high level of selectivity while also admitting a lot of lower SES students.
I suspect retention has more to do with students transferring for financial reasons rather than students being unable to keep up academically. At least, that’s what I’ve observed with students in my friend group starting at Oxford. Quite a few finished up Oxford in 3 semesters and others who took the usual 4 semesters at Oxford plan to graduate from Emory in 3.5 years (including some premeds).