Can science even investigate the supernatural?
Honestly, I don't know.
If we assume that the supernatural does exist, then it seems that science simply cannot investigate it (else why hasn't it). Of course, on the other hand the supernatural may simply not exist.
But I don't see how we can infer from what science has/can investigated about that which it has/can not. It doesn't follow.
I don't really like the term "supernatural" though, because it doesn't impart a lot of
meaning to me. As you question, does it imply an intrinsic property or is it simply assigned as a placeholder for one that is yet to be revealed? I don't think we can scientifically answer that question.
I prefer the term "paranormal".
So lets say we have some supernatural/paranormal activity, how about the
Burning Bush? -- its on fire but its not being consumed. And us scientistist are studying it... but everything we can determine suggests that the bush should be being consumed by the fire and we have no explanation for how it could not be, and yet.. it isn't being consumed. So then what? You just leave it as "we don't know what's happening here". What's the purpose of assigning some property of "supernatural" to it? Esspecially if it doesn't add any meaning to our explanation? At least the term paranormal implies "this isn't normal", i.e. we have no explanation.
Their only source of information about this realm is via a detection system that we know is prone to false positives (the human mind)
Wait, how do we know that's the only source?