Actually, you WOULD have to kill everyone at least once if you want to make all of them immortal, even if you daisy chain it. You can't be resurrected as an immortal if you haven't died. Plus, if you want to bring back people from the past as well as make everyone in the present immortal, you would need more bodies than there are people...
So yeah, the moon eye plan has considerably less hassle and works better overall.