It seems like a silly double standard for only one side to have a burden to prove their claim, but the other gets to claim the negation is true with no burden of proof.
Why is it silly that the claim originally presented should have to present evidence first? The counter-claim only has zero burden of proof so long as the original claim has failed to give any proof of their own.
For example, if you say “2+2 is 4” and my response is “NO IT IS NOT. IT IS 3! I REFUSE TO PROVE IT THOUGH”, not only will I be wrong in a classical arithmetic sense but I have presented no argument for why you ought to believe my new counter claim to your original claim. It would make no sense to believe me without more info in such a case.
You wouldn’t have to present an argument yet, at that stage. I’d think you’re really dumb for needing something like that proven to you, but the initial burden of proof would still be on me. However, when I quickly and easily provide proof that 2 + 2 does equal 4, THEN the burden of proof falls to you to prove your counter-claim.