hate to inject too much gunnerol into this thread. However, there is a legitimate reason a hardcore gunner would want leadership experience.
Quoting from the AOA constitution :
Section 1
Individuals eligible to be elected include:
a. Medical students who have excelled academically and have shown promise of becoming leaders in the profession.
Sounds pretty cut and dried to me. To get AOA, you have to be top 25% and the obvious best way to show promise as a leader is to be a class leader. Unfortunately, politics are involved, such that in theory the valedictorian who is class president could be passed over for AOA membership at some schools.
I'm expecting a response to this post saying "you can get a competitive residency without AOA". Yes, that's true, but statistically AOA is correlated with a very large improvement in one's chances of matching a competitive residency. The odds roughly double.
It could be correlation : AOA members have good medical school grades and good socialization skills, both of which are likely to help someone match a competitive residency. Or it could strong causation if some residency directors put all the AOA applicants into a separate pile on their desk and pitch the rest. It is on the first page of the ERAS app, after all.