> We can't effectively say for certain whether there's a correlation with throwing in a higher power or just group therapy on its own in this circumstance

The higher power is not "thrown in" to AA, it is the critical foundation.

Also the people who are most critical of this concept or most aggressively reject AA based on a perceived religious foundation also tend to be the ones who have the poorest understanding of what the higher power means.

I'll admit it's not an easy concept to grasp. For many/most fellows AA the principles of AA take a good amount of time to begin to understand, and alcoholics sometimes bumble their way in and out for years before this happens. It's a learning process. Pushing back on these principles, especially the existence of one's higher power, correlates highly with continued active drinking. Acceptance of this is the whole point of the program.