"Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied." (*)
The nature of countable infinity is such that there cannot be a last room, and that's fundamental to the fact that new guests can be accommodated in an infinite hotel at will, itself an illustration of the fact that we can count endlessly. But, if there is no last room, it can never be the case that *all* rooms are occupied, hence the whole argument, for how informal, falls apart since inception.
More logical seems to say that, while there can be ideal constructs such as the actual infinities of super-tasks and corresponding "super-numbers", there can be no such thing as the standard countable infinity, as that would be something that at the same time is and is not exhausted, i.e. a self-contradictory notion. Finite infinity is rather just that, the ever unfinished.