The majority of physicians do NOT belong to the AMA.
The AMA protects physicians' collective (financial) interests but may not represent the best interest of the public and may not represent the best interests of physicians who believe that there are more important things than their own financial self-interest.
On the other hand, as a member, you can help influence AMA policy. There are also folks who enjoy "joining" and getting involved in club stuff, local medical association, state association, AMA, etc, going to annual meetings, serving in the house of delegates, traveling and seeing fellow members, etc. I have a friend who did this for about 10 years in a parallel organization for dentists & climbed the ladder to serving as president of the org for a one or two year term. It included some pretty nice parties, too, like a super bowl party at the Ritz, when it coincided with a delegates meeting.
If you read the AMA News (you can get a free e-mail every week with free access to the stories online for 6 mos after publication, that's how I keep up with what they're thinkin' about) you'll see that the AMA froths at the mouth every time a state or a big box moves to expand "nurse in a box" which would, IMHO, go a long way to making access available to many of the 47 million who need episodic care for minor problems.
AMA shills are quoted in the mainstream press every time time the topic comes up. Still, you are correct in stating that most physicians (and I know some twice your age who feel as you do about the AMA) do not believe that the AMA represents their viewpoint.