Agree - to investigate supernatural claims , one would need a 'supernaturalistic' methodology. It should be noted that the lack of such a methodology precludes those who believe in the supernatural claiming anything from what is seen as evidence in the scientific sense which deals with naturalistic probabilities. Any claims that vanishingly unlikely events are supernatural become meaningless as due to the lack of an appropriate methodology, probability is a naturalistic term.
Well this is all now way off topic here, so I'm not intending to continue this further except to point out why you are certainly wrong in the above - look at the highlight of what you say ... you are not actually talking about any supernatural event at all ... you are only talking about humans making claims of knowing anything supernatural.
All you have is what any individual offers as a claim of something happening.
And what you can certainly do with science is to investigate the nature of the person's claim. You can quite easily test whatever evidence or reasoning they offer in support of their claim. The "claim" is not supernatural ... it's the thing that is claimed to be happening which is said to be supernatural ... but before you ever need to try testing for the so-called "supernatural event", you test the veracity of the "claim" itself and the veracity of the individual making the claim.
IOW - all that you have is a claim being made by an ordinary non-supernatural human being. And you can certainly test the likely truth of any spoken claims that people make.
It would be a different matter if the claimant actually produced an event that appeared to be supernatural. But despite vast numbers of people throughout history claiming to describe and show all manner of “supernatural” or “magical” events, whenever those events have ever been available to be investigated, it has always turned out that all such events were/are in fact entirely non-supernatural.