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Whoever is making a claim has the burden of proof.
That may well be a convenient cliched throw-away line to use, however it does not entirely make sense.
For example, if I were to say "there is no God" there can be no burden of proof since one cannot prove the existence of nothing.
Similarly, if I were to say "there is a God", I would have the burden of proof of explaining something that exists over and above the default position of there being nothing. If something is claimed to exist then surely, there must be proof of it.
I think perhaps you have conflated "Whoever is making a claim has the burden of proof" with the old debating standard "Whoever is making a positive assertion has the burden of proof"....which is more logical and makes sense since, "There is a God" ,is a positive assertion and "There is no God" is a refutation of the positive assertion.
We need look no further than modern justice systems whereby a person is deemed innocent unless proven guilty. If you were say, charged with murder ("You murdered that person") the burden of proof is entirely 100% upon the prosecution to prove the charge . As the defendant ("I did not murder that person") you have made a refutation of the positive assertion and you have absolutely no burden to prove such. Indeed, many people have been found not guilty without providing a single piece of evidence in defense.
It would indeed be a world of utter chaos if there were to be a requirement to disprove something that hasn't even been proven in the first place since it would give rise to all sorts of absurd claims that anyone can think of. "Giant green elephants with wings exist.......now go prove that there are no giant green elephants with wings". Or "Donald Trump keeps a boiled egg in his underpants....now go prove that there is no boiled egg in his underpants".
And of course, the all-time classic absurdity: "God exists.....now go prove he doesn't exist".