I don’t think the AA program wants you to rely on the “higher power” getting you to stop drinking. That’s not the point of acknowledging one. The point is to teach you humility, to get you to stop thinking of yourself as the center of the world.
In fact, AA takes the stance that addicts will never completely lose the “obsession to drink”, it is just something they need to learn to live with and not succumb to.
If you are alcoholic, you don’t have the power to stop. That has to come from a power greater than you.