The continuity flaw is a reasonable concern; I have it as well. (I don't want a copy of me to live on; I want to live on, myself.)
However, I do think it's entirely possible to solve, if we're already at the point where uploads are possible. It's harder than the alternative, but theoretically possible.
I've come to believe that the copy will seriously believe it's me, similar to a virtual machine being halted and moved to different hardware, how would "I" know I'm not me? My copy will have my memories, if it's an exact copy it'll have the same ways of thinking, maybe after the transfer it'll remember "Ah, I have a date tonight with my wife, geez I remember this morning she drank all the juice except for a sip, and then put it back in the fridge, that always makes me irrationally angry.".
Sometimes we even feel "out of body", even in our own consciousness, so why would it be so different when we are copied?
One easy way to tell the difference: if you do the full upload non-destructively, such that there's a full uploaded copy that thinks it's you, and your full original biological self, would the latter then happily get disintegrated? (There are people who say yes to that question, which genuinely baffles me.)
The thing I'm talking about, of attempting to build a "move" operation that isn't "copy then delete", is exactly what I meant when I said it seems harder but possible to solve.
Both the copy and the original wouldn't like to be destroyed, would they, it's like volunteering to end one's consciousness.
What if there's hardware to wirelessly combine the minds, and you can experience what your clone^W other-self experiences. A bit like having someone be in e.g. Iceland and facetime you a low quality sight and sounds experience of Iceland... would you then say "Ok, I don't mind having one of my input devices go offline".
> it's entirely possible to solve, if we're already at the point where uploads are possible
The fundamental gating discoveries are all around the nature of consciousness. Is it emergent? Is it empirically detectable? Is it quantum magic? We have inklings around this. But our understanding of it hasn't fundamentally changed since, arguably, Descartes.
However, I do think it's entirely possible to solve, if we're already at the point where uploads are possible. It's harder than the alternative, but theoretically possible.