The claim to the identity is not the issue, the issue is that the person who got copied will die and experience that death.

The existence of a copy does not make the original person to resucitate or otherwise keep perceiving and thinking.

To express it in a bad analogy: you can have a bit by bit backup copy of a harddrive, but when a power surge burns the CPU and the disk, you have to throw both away. You can buy a new CPU and place the backup, but the hardware is different, there is a shutdown moment, and when you power back, the continuity is lost, it's a different entity what gets booted up.

To preserve the consciousness of a person, between the "hardware change" I see no other option to the existence of something like a central repository of consciousness outside both the body and the computer hosting the simulation that gets automatically attached to a particular set of memories/perceptions/experiences (and whatever else defines a consciousness), so when you die, it stores your consciousness and when the simulation is booted up, the continuity is triggered. I find that far fetched.

If consciousness is a very simple emergent phenomenon, is the distinction between one instance and the other still interesting?

In abstract, philosophical terms, it might not be. Even in policy terms it's probably not (other than it might be cheaper to store people in SANs.) In practical, day to day terms, of course it is! My own instance of self doesn't want to cease to exist.