Whilst the logical side of me dictates that the supernatural is implausible, I've had experiences which cannot be explained via conventional means. I don't think we necessarily need explanations for everything but we sure like to have them. What is religion but a way of interpreting and explaining the world in a semi-comprehensible way? Even then, science is still essentially just a way of explaining the universe from our own perspective. Explanations validate your perceptions and ideas not only to yourself, but to those around you. But sometimes the most powerful things in our life are/appear unexplainable.
I think that there are usually explanations for things that people deem supernatural, but people may not find them as satisfying as a supernatural answer. Things like mild halucinations, Sleep Paralysis, cognitive biases and optical illusions. The problem is that these answers can never be known to for sure have caused something, just that they are likely the case. Ironically that while the argument against them is often something along the lines of you can't prove it was a trick of the light/optical illusion that made me see a ghostly figure, the same argument could be applied to saying it is a ghost. At least we know that these tricks can happen to our brains, we don't know ghosts exist.
This is the "absence of evidence is not evidence of absence" reply essentially, correct me if I'm wrong. The problem with this is that statement is not complete, what is should read is "absence of evidence is not evidence of absence unless we would expect there to be evidence". For instance, if I say that a nuclear bomb went off in downtown Denver and you look at downtown Denver and see absolutely zero radiation or destruction, that is an absence of evidence. This does not mean that a nuclear bomb did in fact go off there. We would expect there to be evidence in such an event, and the complete lack of evidence for such a thing is clear evidence that it did not in fact happen. Supernatural claims are, by necessity, claims about the world. If there is evidence that we would expect to see from those claims that is absent then that is, in fact, evidence pointing toward them not existing. It is not really the coin flip you present it as.
I didn't mean to make the absence of evidence argument. I was going more for a lack of absolute proof thing, and how one side has evidence and the other doesn't. Supernatural always have evidence that is more likely something natural. Calling supernatural claims about the world is funny to me, because they are kind of outside of that, thus they are deemed supernatural. However I get your point. I cannot tell if you are arguing for Supernatural phenomena being true, or not.