I think it's misleading to say that natural science a priori excludes the supernatural. This construction is accurate, but only in the sense that married bachelors and round squares are a priori "excluded".
In one sense, "supernatural" is internally contradictory. If we take the natural universe to be everything that exists, then by definition the "supernatural" cannot exist: nothing can exist other than everything that exists. Likewise the term "supernatural cause" is internally contradictory: natural in this sense means "occurring by cause"; a supernatural cause causes something that does not occur by cause.
In another sense, "supernatural" is just another word for "ignorant". If I say something is "supernatural", I'm saying I don't know anything about it. We see this sense often from intelligent design proponents: they attack and undermine evolution to try to establish that we are ignorant of the mechanisms that have produced modern organisms, and then they stop.
In a sense, because natural science can never give us certain truth, we are never "excluding" a "supernatural" explanation: It is always possible that we are ignorant of the true mechanism underlying some phenomenon. It is never possible, however, to prove that some proposed explanation isn't the true explanation without offering a better one. If we think we know, it's impossible to prove that we don't actually know.
Likewise, if we know we don't know, it's impossible to prove that we cannot know. We might convince ourselves that some process is random, but saying "X is random" is itself a natural explanation.
In another sense, "supernatural" means private. Naturalism just means that we understand something by virtue of public faculties, faculties that all ordinary human beings possess and agree on. Those faculties happen to be specifically perception and logical thought, but the specifics follow from the notion of general agreement; we have not simply arbitrarily decided to agree on perception and logic. In this sense, then, you cannot prove the supernatural, because your supernatural knowledge is by definition dependent on some private faculty, i.e. revelation. You can't convince me if the foundation of your argument is unavailable to me.
Every sense of the word "supernatural" is thus excluded from science not by some arbitrary decision, but because all senses are either contradictory or mean epistemically unavailable.