I have read many exorcist articles and books over the years. Basically that can be the rule but there are exceptions. I have read about a few priest in a Rome church where people from Italy or the world go to be delivered. These priests are so attuned to the spirit they can tell right away. They do it almost everyday with lines. Just by the massive experience.
Also remember according to one book people can be both mentally ill and possessed. But you have to have some patience. I remember reading about Smith Wigglesworth He once delivered by just singing Jesus with others. Then another time took days to deliver.
You are quite right.
re Wigglesworth . . . there was a couple having serious problems . . . I forget the nature of them . . . may have been demonic, sexual, what not . . . Wigglesworth for some reason had been spending the night at their house during his campaign. They asked him the last day what they could do . . . and he said to sleep in the bed he slept in without changing anything. They did and the problems were delivered/cured.