Why do all AA members believe AA is the only way to stay sober?
I don't think it is and I'm an AA'er... unless you're the type of alcoholic the book describes. If someone IS that description, I won't say it's impossible but it's not a risk I think is worth taking when your options are: happy sober life vs. dying an alcoholic death. I think you might be using a different (modern day) definition of "sober" than the one
I've been taught by my great-grand sponsor with 51 yrs sobriety that AA subscribes to. Sober = being of sane mind and sound judgment.
Back to your question and my answer though, I should clarify that for some folks it's the only shot they have. Abstinence, for me, did nothing to treat the real problem(s).......and booze wasn't the real problem. Sure, it was A problem but it wasn't the source. Not drinking got me dry but it didn't get me sober.
I don't know anyone "in the program" who's really trying their best to live it that isn't happy AND sober. When you see it working over and over and over and over, it's hard NOT to recommend it, yanno?
The AA book refers, over and over, to "Alcoholics of our type." Not everyone IS an alcoholic of their type. If you are, AA works just about every time (so long as you realize it's more than just "going to meetings"). Nothing else I tried got me truly sober (as I described it).
*and really, it's not AA that works....... it's the power of a God of your understanding doing for you what you couldn't seem to do on your own + helping other alcoholics to get what you've got. AA and the steps are tools we use to get there.