The wording was written by/for Alcoholics Anonymous, and it is used by all the "Twelve-Step Programs" patterned after it. AA was created by members of the Oxford Group, an Evangelical Christian organization for spiritual growth, and is an elaboration of their programming. They expanded a list of 4 practices into a sequence of 12 steps. And while AA adapted itself to addressing non-Christians, their "Big Book" even has a section about how atheists can relate to their program, those roots are very evident. Also pragmatically, they often meet in places like church classrooms or social halls.
Rav Elyashiv was asked about joining AA, and he permitted. As far as I know, the response was never published, but a friend of mine who is a JACS Rabbi has a copy. (JACS is a group for Jewish members of 12 Step Programs and their families; the acronym is for "Jewish Alcoholics, Chemically Dependent Persons and Significant Others".) He not only permitted joining, but even to say "The Lord's Prayer" in the context of their meeting.
Subsequently, debate arose as to how much of the permissive attitude was because of the risk to live involved in addition.
In our terms, there is a dispute as to whether Rav Elyashiv permitted use of the Serenity Prayer, or would only permit the use when someone needs to join such a group to get their life back on track.
As for Maurice Mizrahi's find of a Jewish precedent, this is more akin to the question of using the Lord's Prayer. In both cases, even if the sentiment is inherently Jewish, the fact is the coinage was for another religion. We are prohibited from using others' rituals, even if to worship Hashem Yisbarakh. There was even initial resistance to placing sermons in shul worship over this issue; never mind one of their prayers. Finding a parallel in content isn't enough to permit. Still, Rav Elyashiv might have, anyway.