My understanding of the principle is that you often need a much larger rework of your life to quit an addiction than just changing the habit itself. So much of an addiction is the habits and preconditions that you take before using a drug. For example, first a person might have a lunch break at 12:30, then they'll go and talk to coworker X who also has a lunch break, then they'll take a walk, then they'll start lighting up. All of these little habits have nothing to do with smoking individually, but your brain is already primed to assume 'lunch break -> chat -> walk -> smoke'.

Being willing to alter your deepest held beliefs predisposes you to altering other things about your life, from big things like moving across town to small things like taking a lunch break at a different time.

There's also a social element that I think is hard to reproduce in a secular context. Convincing a bunch of adults to coordinate their schedules is much easier if they believe that they will go to heaven if they do so. Furthermore, you already have a public building as a location to hold meetings. Hell, even the architecture of a lot of churches is awe-inspiring. They're physically designed to be these places where you come to terms with your faults (sins) and begin to deal with them. Of course, that also means that their design is religious, with religious iconography in the stained windows and on the walls.

Charlie Munger has a theory of hour AA works that is pretty convincing. He says that AA uses a number of natural psychological tendencies together to create a "Lollapalooza effect" that helps people to stay stopped, once they quit drinking.