Identity via continuity sounds like the Ship of Theseus problem[0], and it doesn't seem like any kind of a roadblock for anything.

As for being a different being, I personally wouldn't mind. I'm not the same being I was yesterday, or last year, anyway. For instance, over the past few years, I have many parts of my teeth removed and replaced with artificial components. I do not feel worse because of that.

[0] - https://en.wikipedia.org/wiki/Ship_of_Theseus