I was trying to figure out where the cursed part comes into it; I assume it's the switching, that typical circuit analysis just doesn't involve discrete states that are swapped between?
Yep. The state where the capacitor is fully floating relies on non-obvious details. Recall that a (real) transistor has a rated maximum voltage that it can block. But when the (ideal) transistors are opened, there's no defined voltage relationship between (either side of) the floating capacitor and the rest of the circuit; so if analyzed at this level, its not clear if the transistors are within spec or not. You need to at least think about the parasitics to convince yourself that this is sane... or build it with real parts and see that it works.
Yep. The state where the capacitor is fully floating relies on non-obvious details. Recall that a (real) transistor has a rated maximum voltage that it can block. But when the (ideal) transistors are opened, there's no defined voltage relationship between (either side of) the floating capacitor and the rest of the circuit; so if analyzed at this level, its not clear if the transistors are within spec or not. You need to at least think about the parasitics to convince yourself that this is sane... or build it with real parts and see that it works.