Actually, it does reasonably follow that unfalsifiable is illogical, because logical things are open to self correction: logical principles are not the same thing, but something having the property of logic means it isn't self contradictory or otherwise violating those principles that you're doing in suggesting God must be identical with logic, making it unable to be criticized at all.
Un-logical isn't the same as illogical. However, un-logical is perhaps unfalsifiable, so oh well. First Cause With Intent is not self contradictory or otherwise violating those principles. I don't say God is identical with logic. I say logic is his making. No, I can't
Logic is quite falsifiable in terms of application, the principles themselves are axiomatic, but that doesn't make them unfalsifiable, it makes them a practical necessity for cogency. God is not required, far as I'm concerned, for things to make sense, because we don't need to anthropomorphize principles for them to be sensible
But logic itself is unfalsifiable. It is a basis for reasoning, and can be shown to be useful as such, as you indicated, but that is still question begging of a sort. You can't show it to be invalid, not because it is not invalid, but because it is of a sort of thing that cannot be accessed for that purpose.
Where is your evidence or reasoned argument that existence requires a mind behind it rather than that minds apprehend it after the fact of its existence?
Because, if no mind, then you are left with mechanical fact, which I think I already showed to be illogical. Meanwhile, the idea that the mechanical principles we are subject to come from him, because of his very nature, does not at all mean he is mere mechanical fact. You kinda sloughed that thought in. (Furthermore, it is only my opinion that they are the result of his nature --it only makes sense, to me).
Most forceful to me is the acknowledgement, and this too I cannot logically prove, that mere mechanical fact cannot absolutely begin anything new.
Gravity comes to mind as something mechanical in nature that very well could've initiated existence as we understand it
At best, gravity is co-emergent with whatever emerged. Also, cosmologists like Hawking say it too (apparently) began with the big bang, at least gravity as we conceive of it at this point (we still don't know what it is). Meanwhile, to those of us who are a little less advanced in cosmology, gravity is part of a system we might conceive as "what is", if we fail to see "what is" is not First Cause, because it is only a system of principles, that has no self-existence. We want to say they self-perpetuate, and that may be valid (though I doubt it), but it shows nothing about how that system came to be. First Cause is not a system, that operates by laws. That is mere mechanical fact, which I showed to be illogical, I think.