In a way, Columbia already does!
Columbia's School of General Studies is targeted at students without a high school diploma, students with "nontraditional" backgrounds, and students who have taken more than a year off after high school before going to school. I went to Columbia, and while I wasn't in GS, a lot of my friends were.
A large number of GS students are veterans[0]. Of the rest, a surprising number are people with, well, backgrounds that typically disqualify a person from an Ivy League education.
I can't share their stories because they're not mine to share, but I had classmates who were caught (and incarcerated for) drug trafficking and other serious-but-nonviolent crimes[1]. I know this because they told me, and they felt comfortable enough in their academic environment sharing this knowledge. To me, that really says something about not only the admissions process, but also the community and support that they receive as students.
As a non-GS student, I really appreciated this setup. My education would not have been the same had I not been sitting next to these students in our discussion sections for class, or serving on the boards of student organizations with them. And at the same time, they were able to benefit from the structure of a school that was explicitly tailored to students with nontraditional backgrounds.
Columbia could do a lot more to support GS students[3], as well as support incarcerated prisoners who are not students, but this is something they've been doing for decades without any fanfare at all.
Columbia also has a very long history of accepting veterans who were discharged for being gay, even long before Don't Ask, Don't Tell (Stephen Donaldson was a Columbia student before joining the military[4]). IIRC, many of these discharges would have been dishonorable discharges, which in some states carries the same status as being a convicted felon.
[0] Partly because Columbia GS is so proactive about recruiting on military bases, and partly because their yellow ribbon matching is so generous.
[1] I mention nonviolent crimes because that's what (some of) my GS friends shared with me. I don't know of any who were convicted of violent crimes, and I don't know if this is simply selection bias or if there really weren't any.
[3] If anyone reading this has $100MM to spare, the General Studies school could benefit greatly from its own endowment to increase access to financial aid!
[4] https://en.wikipedia.org/wiki/Stephen_Donaldson_%28activist%...