I take issue with the premise that "All rooms are occupied." I think it's an invalid premise, and is an example of English syntax not adequately describing maths, especially when infinities are involved.

The only way that something can be true for all members of an infinite set is if it is part of the definition of that set; if we were to say, "Hilbert's hotel has an infinite number of occupied rooms," then it would be valid. But in that case, there would not be room for new visitors; any room we look at is, by definition, full. There can be no shuffling of guests to higher rooms, because those rooms are full too.

If rooms being occupied is not required by the definition, then the statement that "all rooms are occupied" doesn't make sense for an infinite number of rooms. You can't use "all" in that way.

The apparent paradox is, in my view, an illusion caused by the ambiguity of English words. It's like the fallacy of equivocation (https://en.wikipedia.org/wiki/Equivocation), where the same word is used in different ways to give an apparently nonsensical result. "All" doesn't have its usual meaning when applied to an infinite set.