You know, I suppose that’s one of the very few merits of the “Lunchbox, Lagomorph or Linguine” argument. It is, at least, specific to the particular god in question.
I always find that aspect of a lot of the classic arguments, like those aimed at establishing a “first cause,” really puzzling. You get through the argument and, if you accept its premises and structure, you wind up at “so, there was a first, uncaused cause of some sort or other.” At that point it seems we are all supposed to say, “well, there’s only ONE possibility for what that could be: the most popular god of my particular culture.” For myself, even if I set aside the fact that I think arguments that mostly operate on abstractions rather than on things are useless, I find that it just stops short of that point, and I am left with, “okay, maybe there was some uncaused cause. I guess we should think about whether we could find out what that was.” Though, to be honest, more often my thought at that point is “this sure is a waste of time!”