I don't see why we can't value continuity of consciousness and still go under anesthesia.

We can do that of course. What I tried to fathom with my post was time-consistent self-referencing. We can value our memories, the continuity of experience and even our DNA. But the only reliable marker for agents seems to be a stable utility function. Rational agents with an identical utility function will to some extent converge to exhibit similar behavior and are therefore able to cooperate. We can more consistently identify with our values and goals than with our past and future memories, digitized backups or causal history.