Quite - this is my point. If it has no physical influence, in what sense is it real? In what sense does it exist?
There's no doubt that the concept is real and has physical effects (e.g. on behaviour), but what about its referent?
OTOH, if it does have physical influence, then it is real and physical, so in what sense is it supernatural? What makes it other than natural?
I agree. However, the fact WE can't make empirical use of it doesn't make it not real, only not physical.
Just some thoughts:
We do a lot of abstractions, supposing, guessing, in order to come up with explanations for the physical. True they have to be testable to be proven but we plow ahead without the testing anyway, to come up with some of the notions we have lately in physics. Just saying —we do this, and give credence to some over others.
One of the simplest logics is that everything we see came from something else. While it may be an awkward way to put it, that something else is therefore more REAL than what we see. The fact it is not based on our notion of reality (empirical, or, falsifiable) doesn't mean it is less real, or, at least, less "fact". I'm not going to spend time here defending the idea that everything we see came from something else, except to mention that "Everything that is BECOMING, comes from something that IS."
There are several reasons to believe that God had physical influence, and, philosophical reasons to believe that, in fact, the physical is completely dependent on God for its continued existence and actions. I have philosophical reasons to think that part of the nature of omnipotence is efficiency (and no, not just because it is found in nature —it's not a circular assertion).
So, what is "supernatural" might only mean, "beyond what we consider 'natural'". Like you, I think it was, said, "unusual".