I think the NGT quote goes both ways. This is also a prime example of why making this arbitrary distinction between science and religion is the wrong way to look at the world, and can get very arbitrary with a quickness. After all, the scientific explanation doesn't preclude the religious one.
The problem with your standard is that it presupposes one (but not the other) as extraordinary. And once again, it assumes that this is even an either/or to begin with. I'm fairly skeptical about the idea of demonic possession, but can't definitively rule it out, either. But at the end of the day, something can work even if we don't understand (or are wrong about) why it works.
One claim - The patient is suffering from a psychiatric/psychological malady for which we have no easy explanation or efficacious treatment. The other claim - The patient is host to a demonic entity who is solely responsible for their suffering and whose presence can only be removed by an ordained member of the correct kind (Roman Catholic) of clergy acting according to the will of the correct (Roman Catholic) God. If someone gets exorcised while also receiving appropriate psychological care and they feel better - fine by me.