I don't see the flaw. What's a nonreligious use of theism?

My argument for atheism was from religion's incoherence, not just "from religion." The point is that religions are inevitably incoherent. How is that conclusion affected by the question of whether there are nonreligious roles of theism? Are you saying that once religions are disposed of, there would still be nonreligious theism to consider? What exactly would that be?

This sounds like a semantic point about the choice of labels.