I don't quite agree. Perhaps you'd find it more palatable to say Aristotle's is A proof of the existence of God.
But in order to dispense with God you have to dispense with the logical system under which His existence is a necessary consequence. It can be done, of course.
That's more or less the crux of my last statement. That to deny the existence of a Supreme Being leads to more difficult logical positions, therefore God must exist. You are correct to say that God gave us logic and not the other way round, but that doesn't mean that logic cannot demonstrate the truth of God's existence.
Deep waters indeed. I am inclined to hold that it cannot, although it certainly can strongly suggest it, contingent on the strength of the logical system. A lot of folks smarter than I am have tried it (Leibniz and Godel to name only two). But I don't think that faith would be necessary if it were actually possible to nail the thing down with human logic. A lot of folks smarter than I am disagree... ;-)