The point is it is a valid argument in mathematics similar to what Godel came up with. One of the beauty of mathematics is we question whether it is a man-made system or one found in nature. This should be evidence that God exists in nature.
As for your logical argument:
"It is impossible for God to exist." (Does the atheist make this argument? If so, then they are doing it without any evidence. Thus, it is possible that He exists.)
"It is possible for God to not exist." (Does the atheist make this argument? If so, then Premise 1 is also valid.)
In any formulation we care to discuss, the term GOD is undefined; even if you "operationalize" God, as the authors of the study did, the definition is still not something that is objectively measurable. Any reasoning that goes on regarding this is therefore only speculation. all the authors could do is that, if you operationalize it just so (and they also clearly stated that it was a lot of work to find an operationalization that would actually work), a computer could come up with the same reasoning, and that the pattern of logic holds, given those constraints. It is therefore a MATHEMATICAL PROOF. It is not itself evidence of anything, because you could substitute any other concept into it--as I jokingly suggested, Lephrechuans--and you'd end up getting the same "valid" logical results. The question is, what data from the real world about GOD can put an input into the logical framework? Really, nothing, because the term God is still so poorly defined, defined in ways that are not measurable.
Second, the logic chain Possible::Necessary::Exists is non-sequitur. And it isn't just this particular version of the argument, it's one of the major classical "proofs" of God, the Argument from Necessity. And the critiques of that argument remain the same: possible does not mean does does not mean must does not mean is.
It is possible that humans exist...therefore they MUST exist...therefore they exist...
But really, MUST humans exist? That's an ontological leap of faith...but surprisingly enough, we do have evidence that humans exist.
Then how about them Leprechauns? They might exist, therefore they must exist, therefore they do! The problem is that the only evidence we have for the existence of Leprechauns is that some people claim to have seen them, and some people have written stories about them...which is exactly the same evidence as is claimed for God.