Drawing distinctions between that which the human mind decides is “supernatural” and that which the human mind decides is not super-natural is a task undertaken by fallible humans for their own purposes. We find it a convenient and generally useful exercise in reasoning from our own observations, but have learned over time that many phenomena formerly thought super-natural because they were inexplicable, and also often frightening, partly or largely because inexplicable to our fallible and limited minds, have turned out to be not supernatural at all once we learned how to explain them. Common and obvious example of this shift from supernatural to natural are the eclipses of sun and moon.
"Most of the stories about the Landlord are probably untrue, therefore the rest are probably untrue."
It is true that the plural of anecdotes is not data: but it does not therefore follow that the singular of anecdote is "necessarily false". And the rub there is that most information of a religious kind occurs outside of a laboratory, like the rest of human experience.
Rather an inconvenience for testing -- and what makes it worse are the explicit stories of the sort that "He could do no mighty works there because of their unbelief": implying that a laboratory setting of prayer might defeat its own purpose.
G. K. Chesterton had a good description of this in Chapter 9 of Orthodoxy:
"If you choose to say, "I will believe that Miss Brown called her fiancé a periwinkle or, any other endearing term, if she will repeat the word before seventeen psychologists," then I shall reply, "Very well, if those are your conditions, you will never get the truth, for she certainly will not say it." It is just as unscientific as it is unphilosophical to be surprised that in an unsympathetic atmosphere certain extraordinary sympathies do not arise."
Cheers!