I'm not sure that's what most AA members really believe. I know they believe it for themselves, but having many family members and friends who attend AA I've never heard them say more than AA is the only thing that would have worked for them.
In the Big Book there is a recounting of Rowland H.'s visit to Carl Jung, Jung told him that, "Here and there, once in a while, alcoholics have had what are called vital spiritual experiences. To me these occurrences are phenomena. They appear to be in the nature of huge emotional displacements and rearrangements. Ideas, emotions, and attitudes which were once the guiding forces of the lives of these men are suddenly cast to one side, and a completely new set of conceptions and motives begin to dominate them." What AA has done is essentially converted "here and there, once in a while" into everywhere, all of the time and in so doing has turned a "phenomena" into an everyday occurrence that has been experienced by millions.
You're right and I didn't mean to imply that AA was into self promotion. Shrinks, caregivers, courts and relatives often badger a drinker to "go to AA".