To my knowledge, AA is not a covert attempt at proselytizing for any particular religion. You are perfectly free to invoke His noodly goodness, the Flying Spaghetti Monster, if that's what works for you.
But you also have to understand that the 12-steps of AA are grounded in the personal experience of the founder, "Bill". He had an epiphany, a religious "awakening", that somehow dispelled his desire to drink. He also found that helping other alcoholics helped him maintain his sobriety. And so on, everything he found that helped him was put into the 12 steps. It's not a panacea but it does help a lot of people. They've toned down the religious aspect and made it as vague as possible, but from the very beginning a sincere religious conviction of some sort was required in order to follow in the footsteps of the founder.
There are recovery programs that are totally secular and use a cognitive behavioral therapy approach to find strategies to avoid addictive behavior. There are no "anniversaries" and no guilt; if you fall of the wagon you are encouraged to return and share the experience and try to learn from it. I don't know how it's success rate compares to AA because the anonymity of AA makes it nearly impossible to track such data.