You have a big time logical inconsistency here. Let's say for the moment that via experiment and observation and inference (your terms) that there comes about a scientific finding of a supernatural event. This would be very powerful. There would be an experiment or observation that was repeatable, that via objective reality, all who looked at the evidence could agree on it.
The job of science is to describe how the natural world actually works. The unnatural is not part of science. By definition, this "supernatural effect" becomes a description of the natural.
The result is that God is dethroned. Any mystery is now reduced to being an effect that can be examined in a laboratory, measured, and known. So evidence for a "supernatural" influence actually destroys the supernatural by making it part of the natural, verifiable, measureable part of existence.
No. It is you who persist in a logical inconsistency. You say that science cannot investigate the supernatural when science cannot even define it in the first place. How is science capable of determining what is or is not supernatural without indulging purely subjective semantics? Are you scientifically certain the Law of Gravity is not supernatural? On what basis do you draw this conclusion? Merely because we "understand" it (which we do not understand anyway in regard to its ultimate cause)?
You tell me, if it is possible, how science can decide objectively what is or is not supernatural. So far no evolutionist has been forthcoming without indulging a tautology. "Supernatural is whatever is not natural." Yeah. Right.