But I think it's easy to demonstrate that it would not be like waking up and the "true original" would definitely die in that it would not experience the clone's consciousness
Really? How would you demonstrate that?
Scenario 1: We freeze you, move your body to a different location, and revive you.
Scenario 2: We freeze you, make an atomic copy of the frozen version of you, move the copy to a new location, and revive the copy.
(The reason I want to freeze you rather than simply anesthetize you is that I want there to be no possibility of any physical changes to your body or brain while you are unconscious.)
Unless you are a dualist, you must concede that your external behavior after being revived must be the same in both scenarios. So any experiment we do on you must have the same outcome in both scenarios. So no experiment can determine whether your subjective sense of self is the same or different in the two scenarios.
One of you can't, but two of you might be able to :-) Kind of like how one core can't do real simultaneous multi-tasking, but two cores can.