Infinity is not a number.
There is no room 'n+infinity'. There's just not such a room, anywhere, even in principle. Each guest n can move to room n+1 because the guest there is moving to n+2. This works. That's the point.
And you couldn't, in fact, tell the guests to run down the hallway to find an empty room: There is no empty room until you move everyone.
Infinity is counterintuitive, you say? Well, yes, that's the point of this illustration.
"Infinity" may not be a number, but omega is a number, and it is transfinite. Omega is the first transfinite ordinal.
You are certainly correct, however, that Cantor's hotel does not need to have any rooms with transfinite room numbers.
Not just counterintuitive, but counterfactual and counterphysical. You can't "do it", you can only "model it in the realm of formal mathematics", which by the way proves that this model is not a model of reality, and infinite hotels cannot exist in physics. If you push the physical analysis further, you can get into the speed of light and bosons.
You certainly can't have infinite hotels in physics, but it would be incorrect to assert that we have any good reason to believe that infinite quantities can't exist in nature. For instance, the model of the universe most widely accepted by cosmologists today implies an infinite universe, assuming that our measurements of the curvature of space are correct. And these cosmologists don't believe that these infinities are just a mathematical artifact. Rather, they believe that the universe is likely to be infinitely large, in actuality.
I'm no mathematician, but this seems to be an edge phenomenon that everyone is willing to ignore. I suppose you could just as easily tell the new guests to run down the hallway until they find an empty room at the end; out of sight, out of mind...