[Y]ou’re not buying anything extra for “you”, you’re just creating a new person who happens to share all your memories up to that instant (and presumably your personality traits as well), but then becomes its own entity and lives its own life. A good deal for that new person, of course, but why would “you” care?
He further claims:
[I]t does not grant immortality to the entity actually choosing to undergo the copying process... his consciousness, as an artefact of his brain, stays with his brain; the new body gets its own consciousness, resident in its synthetic brain, which picks up with the exact memories and thoughts the old one had had and then branches off on its own path, but the old one just remains what it was.
But what's the basis for this claim? I don't deny that the copying process would lead to two distinct seats of consciousness, with their respective organic and synthetic physical substrates, but why think that the consciousness of his past self "stays with his brain"? (It's question-begging to hold this on the basis that "his consciousness [is] an artefact of his brain". Each future consciousness is the product of its respective "brain". The question is which, if either, is to be identified with the past self. The above proposal presupposes that the person endures through his organic brain. We're not given any reasons to accept this view.) I agree that it's silly of the character to think he had “a fifty-fifty shot” of ending up in either body. But that's not because the self endures in its organic body. Rather, it's because the self doesn't endure at all.
According to Parfit's reductionism, which I'm sympathetic to, there is no enduring Cartesian Ego of the sort that Kevin's remarks presuppose. I'm having some conscious experiences now, and a future person-stage will have his own experiences and remember mine, but we are two distinct subjects of experience. We exhaust the relevant facts once we state the relations of physical and psychological continuity between the person-stages. There is no "deep further fact" about whether another person-stage is really "me" or not. (Parfit's sorites-type cases help support this conclusion.)
Crucially, on this view, my present consciousness does not "stay" anywhere apart from this very moment. It does not endure into the future. Each moment gives rise to "a new person[-stage] who happens to share all your memories up to that instant". Our everyday persistence consists in no more than the sort of "copying" that Kevin dismisses. Since I value my persistence all the same (even if it merely consists in perdurance, not endurance), I'm led to value "copying" likewise. What matters for our persistence, insofar as our persistence matters at all, are relations of psychological continuity. Future "copies" will be psychologically continuous with our present stage, and this provides us with all the reason we could coherently ask for in order to care about them as a future 'self'. (That's not necessarily to say that it's very much reason.)
Kevin goes on to discuss the tricky problem of double-survival, and ends up accepting a position that sounds much like Parfit's. At least, he suggests that the best way to describe a case of double-survival is to say that neither resulting person is identical to their shared past self (even though each would have been, had it not been for the other survivor). Though I'm not sure whether he goes all the way to accepting that this shows that identity is merely superficial, and isn't what matters for survival. (I'm also not sure whether he means this as a retraction of his earlier claim that the person's consciousness "stays with" their organic brain, and if not, how he would deal with Parfit's more symmetrical split-brain case.)
Anyway, I very much share Kevin's ultimate conclusion about the topic: "Good fun." Perhaps I should conclude by returning to his original question: given that survival doesn't "buy anything extra" for your momentary self, "why would “you[-now]” care?"