Sobriety, however it comes to someone -- alcoholic or not -- is always by the grace of God.
But God can only do for us, what we are willing to accept and use. God can use more than one method to bring us to a point where we are willing to learn and willing to try whatever we can to escape the suffering of alcoholism.
Everyone has a different experience how they were given the miracle and gift of sobriety. This in no way lessens anyone else's experience of God and a plan for sobriety.
I would not like to see AA limited to only Christians or even believers. Many who are sick and suffering don't believe in anything until they come to a 12-Step program or some other God-given program and accept the help. Then they can, perhaps, recognize the Divine in their own way.
For instance, someone who says AA "didn't work" for them but finding Jesus or Buddha or Jehovah did work to create sobriety and a new life--to me that is the same thing in different words.
Turning ones will and life over to the care of a Higher Power, means surrendering to God (or Christ, or any other expression of the Divine) and stopping trying to work it out on the ego level.
It's like surrendering to the sun for light instead of insisting on using a candle & trying to do something all by yourself. (And if someone can do that, do it alone, that's great, too.)
Sobriety is a miracle and a gift. But we have to accept our gifts since God will never take away our free will. (By the way, free will and suffering may be the shortest distance to finding a lasting spiritual path.)
A lot of it is semantics (words) -- i.e., calling the same phenomenon and spiritual experience by different names. There is really no difference between "turning your life over to a Higher Power" and being "born again" or "saved" or "enlightened" etc. The point is everyone can surrender to the divine powers.
God would not turn anyone aside because they didn't call him by the correct name. God is pure love and joy and willingly helps "his children" to help each other and themselves through their experience and sharing.