athletes

There is not enough time in the day/week to do both

Greg McElroy was an SAE. Again, though, you have to have lots of self-discipline.

Do athletes choose to not rush or do their coaches not allow them?

There are a few but, usually the team itself is typically like it's own Sorority or Fraternity. UA also has it's own study hall that is required for athletes. Then, there is practice time and workouts. Unlike sororities and fraternities, athletes are required to go to class. There are secret people/students that are given the athletes schedule and show up to classes to check and see if the athletes are in attendance. If they are absent, there are huge consequences with coaches and athletic department. All of this does not leave much time for Greek requirements. Again, your TEAM is your own fraternity or sorority so, why would you pay for one?

They don't need Greek Letters to make themselves feel popular on campus, or to get laid.

Same reason a lot of tough majors don't. Not enough time/don't want to lose focus.

I mean the time thing, but even if they had extra time (which most do, they are in b/s majors and get study guides which look like 90% of the actual test)...it's that they are too good for sororitys/frats.

Maybe it's oldrow, but not to burst anyone's bubble here, but sororitys/frats are looked down upon. No, not the whole party scene. It's that you are seen as losers who can't make friends, that's why you go Greek.

While it may be cook to show your letters here (Auburn, Ole Miss etc), outside of these two states you are just viewed as losers.

Actual top tier is athlete's (football, not track lol) and your pre-meds, smart people.
Then you get into ranking the Greeks. Then you have GDI's again in the bottom. So, basically GDI's are split in top and bottom.

Most athletes don't have time to join. Some of the coaches actively discourage joining because they want their team to bond to each other. A handful of athletes join, but very few. Often they have to be excused from a lot of activities and don't bond with other members as well as non-athletes.