That's certainly the common wisdom, and I would consider it a plausible answer that there simply aren't any Holy Men on earth with the same order of magnitude of fame as George Washington or Abraham Lincoln. In which case the scenario could drag on for a long time until a ruinous technological solution is found.
Unless you are changing the terms of the OP, nothing requires that a man of faith be well-known to the world in order to be able to exorcise a demon.