Short answer, yes that's a good way to think about it. The Hilbert Hotel paradox is closely related to the B-T paradox, it only adds a few more complications to make a stronger statement.

The key step in the B-T paradox is setting up a two-dimenional free group in just three-dimensional rotations. This is an infinite discrete set, a sort of analogue of the integers (which is the one-dimensional free group). In the integers it is no surprise (or maybe it is) that you can take the positive integers and shift them all to the right by one unit and then suddenly you have made a "hole" from apparently nothing (the Hilbert's hotel paradox). One worrisome part of this is that the "rooms" must get infinitely far away, but B-T exploits the fact that by using irrational rotations you an wrap the whole construction into the space taken up by a single sphere.

Add just a little more mathematical mumbo jumbo and you have a two-dimensional free group, which is a hotel such that every room is as big as all the rest of the rooms and you can set up the hotel so that each room is as big as all the rest of the rooms, and then you can do magic such that a move to the left actually leaves as much space as the whole hotel. And since it's all wrapped up in a ball it doesn't even run off to infinity to do this.

You do need the rooms to be points with zero volume, though; otherwise even with one-atom rooms you still need to shove the entire infinite hotel into a finite space and you will run out of room.