When we study (some) natural things, the question of their existence is established by a different power than the theoretical exposition we give of them. The existence of birds is established by simply seeing them while ornithology involves classificatory schemes, theoretical structures, etc. Because of this a problem or paradox or unthinkable conclusion in the theoretical structure of ornithology does not count as evidence that birds do not exist.
Natural theology is not like this. The existence of the being it studies is established by the same sort of theoretical inference as we use to explicate properties. Because of this, a problem or paradox or unthinkable conclusion in the conclusions of natural theology does (or at least can) count as evidence that the God of the Philosophers does not exist. Said another way, while a strange or unthinkable conclusion in ornithology does not count as a reductio ad absurdum against the existence of birds, it does in natural theology.
But there are many unthinkable conclusions in natural theology (viz. a subject identical to its operation, a being that is free but unable to be determined by anything, a being that must be understood by both concrete and abstract terms, a being whose essence includes the notion of existence, etc.)
Therefore, the unthinkable conclusions of natural theology count as evidence that the God of the Philosophers does not exist.
(It might be better to make the argument turn on the fact that natural theology has no standard by which to distinguish a mere paradoxical conclusion from a reductio ad absurdum against the claim that God exists; and it seems to set the level of proof absurdly high to say we must reach a formal logical contradiction before we have a reductio ad absurdum.)