[nwallin]

Isn't that just because Hilbert's Hotel is a property of the natural numbers (well ordered) while Banach-Tarski is a property of the reals? (not well ordered without AoC) We can split the natural numbers into odd and even groups by starting with 1 and iterating on the odds, and starting with 2 and iterating on the evens. But because the reals are not well ordered, the step in Banach-Tarski where we pick an arbitrary point that hasn't already been grouped into a set is impossible.

The natural numbers (and therefore Hilbert's Hotel) provide a natural way to say "whatever, just pick one" but we need to invoke the well-ordering theorem (which is equivalent to the Axiom of Choice) make the same "whatever, just pick one" statement about the reals. (and therefore Banach-Tarski)