Sims are actually born with a time of death.
This is
not true. Elder sims are given a percentage chance to die each day that is based on the number of days that have passed since becoming elder. This percentage is marginally affected by things like Vegetarianism and the Marathon Runner challenge. This is information that comes from someone who designs game mods, and has seen the actual game code.
Likewise, my experience with death by old age is dramatically different. I often forget to save, and sometimes a significant amount of time passes between saves. One time, I had an elder die at well over 100 days old, and had a game issue that forced me to reset without saving. That sim later died somewhere around 80 days. Another time, I had a sim die very quickly after becoming an elder (in the high 70's). In this game, the sim in question died at work and the Grim Reaper never showed up to claim him. So I reset the game and, assuming that he'd die around the same time, I had him retire. He lived to well over 100 the next time.