The author's argument seems to be a practical one and two-part: 1) without death, there's nothing to motivate us to live life well and 2) unless we live life well, there's no point in living.
I just disagree with both postulates, and that's fine. The author can go on thinking that life needs to be something specific in order for it to be desirable. I myself like being productive. I also like eating fast food every once in a while. I think I'd be able to go on living (with some happiness to boot) if I never had another productive day or another McD's burger ever again.
Life can be its own end. If we manage to end death by aging, someday there will be children who have never known another world, and they'll marvel at all the death-centric thinking that permeated the societies of their past.
I think the point is a bit more nuanced and has to do with the authors conception of the self. He argues that even if you got immortality and lived a great life at some point You would stop being You so you might as well have died anyways. I think it’s a bit silly. But if you believe that enough alteration of the self results in its death, a sort of Self of Theseus, then I think it’s a consistent opinion.
> His argument is precise: the desires that give you reason to keep living (he calls them categorical desires) would either eventually exhaust themselves, leaving you in a state of "boredom, indifference and coldness", or they'd evolve so completely that you'd become a different person anyway. Either way, the You that wanted immortality doesn't get it. You just die from a lack of Self rather than through physical mortality.