So your issue is that a copy of you is not you? And you would treat star trek-like transporter beams as murder?

Nothing so melodramatic, but I wouldn't use them. UNLESS they were in fact manipulating my wave function directly somehow causing my amplitude to increase in one place and decrease in another. Probably not what the screenplay writers had in mind, though.

But you are OK with a gradual replacement of your brain, just not with a complete one?

Maybe even a complete one eventually. If the vast majority of my cognition has migrated to the synthetic regions, it may not seem as much of a loss when parts of the biological brain break down and have to be replaced. Hard to speak on behalf of my future self with only what I know now. This is speculation.

How fast would the parts need to be replaced to preserve this "experience of continuity"?

This is an empirical question that could be answered if/when it becomes possible perform for real the thought experiment I described (the second one, with the blank brain being attached to the existing brain).

Basically, what I am unclear on is whether your issue is continuity of experience or cloning.

Continuity. I'm not opposed to non-destructive copies of me, but I don't see them as inherently beneficial to me either.