Because if the supernatural exists, then the laws of nature don't apply, nor do the rules of logic.
If the supernatural exists, it would have to have some influence on the material world. Even if the supernatural didn't conform to logic or nature,
it's impact on the material universe would and thus that impact could be studied. If not, then saying "if the supernatural exists, then the laws of nature don't apply, nor do the rules of logic" is irrelevant gibberish. Again, the burden of proof is on the claimant making the positive claim. Arguing that the supernatural exists
is a positive claim.
No, that's not how burden of proof works. Neither position is automatically subject to it, it's on whomever makes positive claim.
Not exactly, those proclaiming the existence of the supernatural and/or a god are making the positive claim. Both theists and non-theists accept the material world, that planets, rocks, people, molecules and atoms exist and are subject to scientific inquiry via physical means. The theist/supernaturalist proposes an
additional component above and beyond the material world, and thus the onus
does fall on them to prove the existence of god(s) or the supernatural.
The theist can always shrug and say it's the unpredictable, unverifiable whim of their deity...
You seem to think that's an advantage, or at least an excuse.
I'm unsure as to what you mean here. Are you saying I think it's an advantage for the theist in that they can fall back on the ambiguous/unverifiable/inscrutable nature of their god to avoid analysis? Well, it is an advantage for the theist when it comes to actually discussing their beliefs and substantiating it.
I posted a similar thing in another thread earlier: can you imagine an atheist using such arguments to substantiate their non-thiesm?