It isn't by necessity a religious program, though I freely acknowledge its theistic roots, and the fact that many are religious and do rely on deity as higher power.
But the reason these people were capable of this bravery is stated in the article and is specifically not their piety - it's their honesty.
"Step 4: Make a searching and fearless moral inventory of ourselves"
"Step 10: Continued to take personal inventory and when we were wrong promptly admitted it."
The most important lesson to be learned in AA has nothing to do with God and everything to do with addressing falsehoods - the lies people tell themselves and others to justify their behavior and to excuse their actions.
Through time, habit, and conscious effort and will, these people have primed their minds to be willing to accept a fundamentally difficult truth - that what we think and what we feel can be false. That the things we tell ourselves, the things we tell others, and the things we do can all be wrong.
We all have a responsibility to face those truths with courage and transparency. We have a responsibility to own our flaws and mistakes and make amends where possible. That is the guiding truth of AA. It all started with God, but it ends with the individual, and how they face those truths.
From a quick ddg -
Source: https://aaagnostica.org/2020/03/29/staying-sober-without-god-practical-step-eleven/
You don't have to substitute "God" directly in the steps to make them work for you. There are plenty of ways to use the ideas of the program without being limited by its theistic roots.
Of course AA works because it serves as group therapy. That should be fairly obvious to anyone who's ever heard of the concept. But the most important step in any therapeutic approach is acknowledging hard truths. That is the most important part of AA, as well.
Half the steps are devoted to honestly acknowledging our flaws and mistakes, owning them, addressing them, and making amends wherever possible. That is what these pardon refusers did here, and the world would be a better place if more people had their courage.