That's not a proof by counter-example, that's a proof by reduction to absurdity, and you don't need to make a case for what you're disproving, you just need to

That's not a proof by counter-example, that's a proof by reduction to absurdity, and you don't need to make a case for what you're disproving, you just need to show it leads to something false. Whatever argument anyone has towards it being true is pointless and doesn't need to be mentioned. A proof by counter-example can get even more terse, you simply have to construct an object which breaks a hypothesis. You don't need to consider the incorrect arguments either. Sometimes these proofs are only a sentence long.