Isn't continuity just an implementation detail? Suppose your brain was replaced a bit at a time with mechanical hardware, the end result is an uploaded mind while maintaining continuity.
I admit that this is a troubling problem with the position that I stated, but I don't think it's a complete takedown.
The easiest rebuttal would be to simply say that continuity is not a mere implementation detail. If you give up continuity, you can make a copy without altering the original, you just have to read it.
But if you need to ensure continuity you have to alter the original. This seems to me a very fundamental part of the process, making it qualitatively different.
