> I'm really struggling to square up my understanding of how the former problem works with my intuition that if every room is in an occupied state, you can't magic up some in an unoccupied state.

That's the problem with infinites. Consider the number of all positive integers. Let's call it a1.

Now consider the number of all even positive integers. Let's call it a2.

Each even integer can be directly mapped to an integer and the other way around by multiplying or dividing by 2, right? All integers can be multiplied by 2, and all even integers can be divided by 2.

So a1 = a2, the number of integers is the same as the number of even integers, even though you'd intuit the number of integers to be twice the number of even integers.

Just as it does when reaching the infinitely small (0.999… is precisely equal to 1), intuition breaks down when reaching into the infinitely big.