Yea, there is the nomination-by-your-peers element that can be viewed as a popularity contest. However, school-specific secondary process that includes the selection committee's review of your clerkship evaluations, records of community services, and interviews with faculty and current GHHS members that make up the final decision is where it isn't a popularity contest. It is school specific and was the reason I said "at my school."
Specialty and/or program dependent, but residency programs may look at everything, including your MCAT and undergrad college transcript. Program residenct ranking process is often by scoring each candidate. You want to be near the top. While not a defining factor alone, GHHS can enhance your application. Hell, I was an Eagle Scout in high school and it came up on a number of my residency interviews regarding leadership aspects.
Edit: forgot to mention that at my T5 program, residents have a track record of AOA and/or GHHS.