AOA is always a coveted status symbol, but I'm not sure that it is the critical factor in fueling competitiveness among classmates. At many places AOA selection becomes a numbers game with a small input based on outside activity such as leadership and activities. My impression is that it really commes down to grades though.
No matter what medical school you go to, the top group of students in the class will have a degree of inherent competitiveness. They get to that top position by being smart and working hard. If a school has class rankings (and what school doesn't provide some sort of class rank to residency program directors) I don't see how adding AOA is much different.
AOA basically means that a student is at the top of their class, and a high class rank should mean nearly the same thing under most situations.
Benefit: AOA is instantly recognizable as a "gold star." Just a simple statement that Student X is ranked 14th in his class of 100 just doesn't seem to carry the same flair that AOA can bring to an application.
I always wondered how the situation would be for students whose school has no AOA. (and the opinions of others on this question might just be the support you need to convince your school to start a chapter)
Are students at a disadvantage if they have to say "no I'm not AOA, but that's because my school doesn't have an AOA chapter." ???