So maybe all the religious dogma is a placebo and the real help is the fact that it puts alcoholics in an environment that both supports them and discourages them from relapsing, but their program does indeed 'work.'
That's pretty much my take on things. I've come to see alcoholism as an attitude, at least in part -- a certain way of looking at the world and one's place in it. I can't say whether this is symptom or cause, and at some point, it may not even matter, but one thing I have noticed is that when some people quit drinking, instead of an improvement in their attitude, what happens is it gets worse, at least for a while. There doesn't seem to be much more to it than simply toughing it out. It's not merely the insight that can be provided by a person who has been through that; it's that some things are just harder to do on your own.
Think of all the people who go to a gym to lose weight. I mean, sure, the equipment is nice and all, but lots of those folks have equipment at home that would serve just as well if they'd use it. Thing is, they don't. When it comes right down to it then, the main thing is that most people find it easier to push themselves to a greater level of effort when they're in a room full of people who are all committed to doing the same.
I do, however, have to question your use of the phrase "religious dogma", taking that to refer to "a body of doctrines formally or authoritatively confirmed". The AA approach is, to my way of thinking, just about the opposite of that. Phraseology such as "higher power" and "God as we understand Him" were chosen explicitly in order to abdicate to the individual the authority to make those determinations, and the book declares itself to be "suggestive only" and goes on to say "we realize we know only a little". That doesn't mean there aren't plenty of people in AA who take a dogmatic approach to such things; there are. But AA itself does not take a dogmatic approach to religious beliefs; its stance is: you're on your own there.