[cvoss]

> you didn't really prove every item gets counted in finite time

The proof that the adversary cannot name such an item is logically the same as a proof that every item gets counted. This is a fundamental logical truth, namely de Morgan's law for universal/existential quantifiers: (not exists x such that P(x)) is the same as (forall x, not P(x))

> Why that requirement has any relevance here

'A counting scheme' is an intuitive way of providing a 'one-to-one correspondence to the natural numbers,' which is used in the technical definition of countability.