Argh! No, damn it, I live in the set theory that really does have all the subsets, with no mysteriously missing subsets or mysterious extra numbers,

Indeed, I think it's somewhat unclear what is meant here. The speaker attempts to relate it to physics, referring to the idea that we appear to live in continuous space... but how does the speaker propose to rule out infinitesimals and other nonstandard entities? (The speaker only seems to indicate horror about devils living in the cracks.) Or, for that matter, countable models of the reals, as someone already mentioned. This isn't directly related to the question of what set theory is true, what set theory we live in, etc... (Perhaps the speaker's intention in this line was to assume that we live in a Tegmark multiverse, so that we literally do live in some set theory?)

Instead, I think the speaker should have argued that we can refer to this state of affairs, not that it must be the true state of affairs. To give another example, I'm not at all convinced that time must correspond to the standard model of the natural numbers (I'm not even sure it doesn't loop back upon itself eventually, when it comes down to it, though I agree that causal models disallow this and I find it improbable for that reason). Yet, I'm (relatively) happy to say that we can at least refer to this as a possible state of affairs. (Perhaps with a qualifier: "Unless peano arithmetic turns out to be inconsistent...")