There's nothing fundamental we're missing: the reality beneath our intuitive understanding of personal identity and continuity is just a lot messier than our intuitions allow for. Nondiverging copies are, in some sense, the same person, but they're also completely useless: the instant you "wake them up", they diverge and become different people. Destructive "moving" preserves personal identity, but is goddamn creepy because it provides no way for the "moved" person to verify that they remain the same before-and-after. The "continuous stream of consciousness" model is intuitive and lets us detect when something in our heads changes (you can feel yourself going from "sober" to "drunk" while awake in a very different way than if I just injected you with booze while you slept... For Science, of course).