One way to look at this is to note ask what exactly it meant when you said "a sphere of volume". In the real world it's ultimately impossible to imagine spaces

One way to look at this is to note ask what exactly it meant when you said "a sphere of volume". In the real world it's ultimately impossible to imagine spaces which don't have volume of some kind or another, but in mathematics we often study models of the world too sparse to even have "measurability". These are obviously aphysical, but it turns out that when one tries to construct reality via set theory you have to go through a zone of "funky aphysical things" before you reject them all and consider only measurable things. BT exists just before we make that last transition and abuses the fact that the constructions we've made so far are not required to have any reasonable sense of "volume" whatsoever. All of the "infinity" bits are sort of a red herring as there's no reasonable way to think about taking physical things and cutting them into aphysical things. Instead, this all arises from taking incredibly aphysical things (raw, theoretical sets) and building them up until they feel physical. If you lay out the constraint that "cuts" must divide measurable things into measurable things, which is a reasonable expectation we have of physical items, then BT will vanish.