Also, you run the risk of seeming one-dimensional. Pretty much everybody can join a greek organization. It's not all that unique, though I'm sure the experience you detail in your PS is. I just wouldn't want an adcomm to read my file and think, okay, so this person was in a fraternity and held some offices. Big deal.
You're right, being in a fraternity is not necessarily unique. But then again, neither is a lot of the other backgrounds I've read on here. I'm not disagreeing with what you're saying, I'm just saying the idea seems so bass ackwards to me. And I don't think the point to take away from my PS would be that I was trying to make a "big deal" out of the fact that I held a position, but rather the experiences that shaped me into who I am. Hence, a "personal" statement.
TITCR. If you think it will be your best PS topic, I say go for it. I don't see anything wrong with writing about specific experiences within your fraternity that exemplify your character and leadership and demonstrate how your experiences helped shape who you are today. Avoid sounding stereotypical/bro-y: use "fraternity" instead of "frat", use language that is intelligible to independents (e.g.: "joining a fraternity" instead of "pledging a house"), and avoid language that induces images of alcoholism and hazing.
I agree, that anyone can join a fraternity (key article: "a") is largely irrelevant. I submit that anyone can do most of the stuff people write about in their PSs, excluding PSs about inherent characteristics (race, sexuality, place of birth, etc). Anyone can
save puppies. Whether you're writing about saving puppies or serving as a leader of your fraternity, law schools will be most impressed by essays that are genuine, well-written, and convey personality traits that law schools find desirable and attractive, such as character, ambition, leadership, and compassion.