I would say that the entire value and interest of the Banach Tarski paradox lies in the fact that you are restricted to seemingly volume preserving transformations, thus proving that volume cannot be meaningfully defined on all sets of points in space. This is not an obvious fact, and in fact we can come up with definitions of volume that work on very large collections of sets of points in space. Without that you just get some very basic set theory, which is a lot less interesting and surprising than Banach Tarski. I wish the author had only claimed to be explaining basic set theory, instead of explaining Banach Tarski, in which case it would have been quite a good explanation.