Completely off topic, it turns out that it's barely possible to confirm that Hilbert ever told the story of Hilbert's hotel. He never wrote about it and evidently only mentioned it once. It was either repeated or invented by George Gamow in his pop-math book, One, Two, Three, Infinity. A historian of science tracked down the fact that Hilbert mentioned the story in a public lecture in 1923, never mentioned it again, and neither did anyone else till Gamow's book twenty years later.
The Hilbert hotel story is a lot like rubber sheet and bowling ball gravity. It's just a nontechnical popularization of an idea, not the idea itself.
Galileo noted that we can put the natural numbers 1, 2, 3, ... into one-to-one correspondence with the perfect squares 1, 4, 9, ... but never went further with the idea. Today it's taken as one of the definitions of infinite sets that they can be put into bijection with a proper subject of themselves.
There are of course no infinite hotels, infinite supplies of guests, etc. It often confuses the issue to talk about Hilbert's hotel as if it's a mathematical argument; when in fact it's no more so than a bowling ball deforming a rubber sheet is an argument in relativistic gravity. Both stories are just popularizations for the tourists and are not to be taken literally.
If it were me, I'd abolish the story of Hilbert's hotel in favor of more precise mathematical arguments. The historical record is clear that Hilbert never took the story seriously, though he of course was extremely well versed in the mathematics of infinity.
OP, if I have the infinite set 1, 2, 3, 4, ... and I add 0 so the set, haven't I just added to infinity?