"Religious folk are perfectly happy for all gods OTHER than their own selection not to exist."

That does generally seems to be how "religious folk" approach the issue, at least so far as the Christians I know are concerned. One ironic thing about that, is: Christians (conservatives, at least) acknowledge the same God presented in the Old Testament. But the Old Testament is by no means consistently monotheistic. It teaches that Yahweh created everything, which perhaps implies that the other gods aren't really worth calling gods. It teaches that you shouldn't worship any of the others, but doesn't really say they don't exist. There are various instances in which "Show-downs" between Yahweh and some other god occur, which serve to prove that "my God can whup your god", but which tend if anything to imply that the other god exists.
There may be claims in places that only Yahweh exists, but the O.T. books aren't consistent about it.