You mentioned the difficulty of proving non-existence. Specifically,
A lack of evidence does not prove non-existence.
This speaks to the problem of "proving a negative" which we encounter in popular cultural slogans such as "you can't prove a negative." I am making the point that we actually can (weakly) prove a negative under certain conditions. We can even "prove"
via lack of positive evidence that Bigfoot does not exist
such that the burden of proof rests with those who claim otherwise.
This is a more powerful refutation that the formal pattern of presumption you invoke (i.e., one who asserts positively, must prove positively), because the stance I propose involves scientific inquiry into this domain.
It's not just that pro-Bigfooters haven't proved their claim, but that we also have good evidence to conclude otherwise. It's not just that Bigfoot must produce some positive evidence, but that Bigfoot must deal with the evidence we have to the contrary (failed searches, debunked evidence, suspicious conspiracies needed to float the plausibility of large hominids meandering about North American forests in an age of ubiquitous surveillance, etc.).
"Proof" of Bigfoot's existence would be easy (and virtually 100%) if there was a body or remains that could be tested.
I agree. I am making the point that you are right. I am supporting your flank with cavalry.
Tally ho!
What's wrong with what I said about not buying into the supernatural theories due to the fact that the very definition of supernatural means it cannot be subjected to the scientific method in any conclusive way, while anything natural can be?
Literally everything. What you said is a paradigm case of circular reasoning. By the same method I can prove that God exists, because the Bible tells me so and that I can trust the Bible because it is the word of God.
There is no scientific model for discovering magical beings
That's the problem. That's exactly the problem. Science begins with assumptions, not pure observations. It assumes that all that is real is what you can touch and what you can feel and that the only evidence which is respectable come from this domain. Naturalism is non-falsifiable by science, because it is a pre-commitment of science. Science can only falsify claims which are downstream of its assumptions.
there is a profusion of models for animals being discovered to exist that were previously thought to be only legendary or extinct.
Models? Such as... ...what?
Examples? Yes. Animals fit in the container of naturalism and thus claims about them are subject to falsification by scientific means. Gods and ghosts don't fit in this container, by definition, (i.e., they are SUPER-natural, outside of nature) so they cannot be falsified by scientific means. We falsify the Bigfoot theory scientifically. We cannot scientifically disprove the God hypothesis, however, in a non-circular manner. Rather, we have to shift to a meta-discussion about the virtues of our explanatory backgrounds (naturalism vs. supernaturalism).
You'd need an extraordinary amount & quality of evidence to prove something exists and is supernatural
What evidence? If it is supernatural, it is OUTSIDE of nature. The only evidence the scientist accepts is INSIDE of nature.
At most, science can naturalize what was formerly believed to be supernatural. If we can find natural evidence of a phenomena, it is a natural phenomena.
Suppose, for example, we were talking about ESP, right? On a supernatural account of ESP something entirely outside nature communicates information to the a person with ESP. If so, we cannot scientifically prove or disprove that which science does not observe and which it assumes does not exist. On the other hand, suppose we found that brains emit an odd frequency of electro-magnetic radiation which enough neurons fire in a given way and that these waves can be sensed by people brains structured with certain neuronal structures. We have scientifically proved the existence of ESP at the cost of naturalizing it. That is, it is not immaterial ghosts doing the work, but testable natural processes. The supernatural is thus deflated to the natural in this example. That's all science can do. It's a hammer. It drives nails.
Complaining that supernatural explanations do not pass scientific muster is like complaining that squares are not circles or that you can't fit the square peg in the round hole.
My point is: the model for discovering new, unknown species with the conclusion that these animals are real is redundant - it's happened many times (in other words, such discoveries are as common as mud... well, maybe not THAT common, but it's happened enough to not be some fantastical idea),
Yes, within the container of science, naturalistic claims can be falsified or verified (although not absolutely because the realm is empirical) naturalistically. This is the easy (non-circular) case.
while discovering "magic" or supernatural things has never been done at any conclusive level
It's not that it hasn't been done, it's that it can't be done. Full stop.
and even things that appear to be supernatural may not be able to be proved to be such (whereas proving say Pandas - previously thought to be mythical - are real animals; all you have to do is find one and capture it or test a sample of its biology.)
And in so doing you will naturalize it. Science cannot prove (or disprove) the supernatural. Rather it can only naturalize what was previously thought to be non-existent.
Consequently, you cannot fault a supernaturalist for failing to meet non-supernatural criteria. However, you can press the supernaturalist to explain why we should accept his background in the first place (i.e., we have to climb up a level of abstraction).