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.

That's not what I'm claiming. I'm saying the claim AND counter-claim should provide evidence/proof before either one is accepted. Blindly believing not B because you can't prove B is just as bad in my opinion as believing B itself with no proof.

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.

A lack of evidence or proof for some claim B is not sufficient proof for not B. It doesn't really matter what claim we assign to B here.

For example, you might not have evidence/proof that it will rain today (i.e. B is the statement "it will rain today"), that doesn't give you sufficient evidence/proof to now claim that it will not rain today. You just don't know either way.