I graduated from one of those univs where I've seen a number of my classmates get hired with low mins or even straight out of school.
It seems to me that the guys that do get these opportunities (they do seem like a lot because I know a number of them, but actually are only a small percentage of the entire school body) are in top of their class, very high GPAs, leaders in clubs, and/or working part time, etc. they obviously worked very hard not just with school but extracurriculars, flying, work, all simultaneously.
I know some of them personally, and to top if off, they're great guys. The regionals know this already, because, and this is what seems to be the deciding factor:

Internships. My school offers internships to many airlines. These are very competitive and high mins (for ex., must have a minimum 3.0 and their commercial/multi to even apply, but most competitive applicants have like 3.7 and their CFIs as well) and so only a few that apply actually get it. But completing the internship successfully guarantees an interview later on or slashed minimums. Notice I said "successfully", as internships obviously aren't an easy shortcut out.. not only did you have to work your butt off for 3years to even get the internship, you obviously have to work hard at impressing everyone at the company for the duration of a semester, or however long the intership lasts.

That's my observation. I wish I had done the same, but 1. my grades didnt cut it and 2. I downplayed the importance of internships when I was younger and stupider.

Oh yeah, as far as graduates from flight schools like ATP, they have formal agreements between them and a number of regionals that guarantee interviews at like 500 TT or whatever.