Suppose I'm destructively uploaded. Let's assume also that my consciousness is destroyed, a new consciousness is created for the upload, and there is no continuity. The upload of me will continue to think what I would've thought, feel what I would've felt, choose what I would've chosen, and generally optimize the world in the way I would've. The only thing it would lack is my "original consciousness", which doesn't seem to have any observable effect in the world. Saying that there's no conscious continuity doesn't seem meaningful. The only actual observation we could make is that the process I tend to label "me" is made of different matter, but who cares?

I think a lot of the confusion about this is treating consciousness as an actual entity separate from the process it's identified with, which somehow fails to transfer over. I think that if consciousness is something worth talking about, then it's a property of that process itself, and is agnostic toward what's running the process.