I experienced similar cognitive dissonance too, AnotherLife, when I was going to AA meetings. Most of the reasons were the same as yours, except for the belief in God. I had a strong belief there, but the God of my understanding wasn't anything like the AA one.
I had not yet heard of any other types of recovery tools at that point, but the dissonance was very strong, I simply could not see any way forward there for me, even though I persisted for a while at a few different groups. Despite the naysaying and prophesies of doom, I never did return to AA, AND I never drank again.
It wasn't at all a question of whether the AA modality works, it's whether it would ever work for me. And for me, the tail chasing would have been to continue to attend these meetings.
At the very root of this, regardless of the approach, is the necessity of only one step. Stop drinking, and never do it again. There are many ways to achieve this, to support this single step. I hope you find one that makes sense for you.