For example, suppose that you live on average 80 years. If some event which causes you near-infinite disutility happens every 80.01 years, you should ignore it; if it happens every 79.99 years, then preventing it becomes the entire focus of your existence.

That only applies if you're going to live exactly 80 years. If your lifespan is some distribution which is centered around 80 years, you should gradually stop caring as the frequency of the event goes up past 80 years, the amount by which you've stopped caring depending on the distribution. It doesn't go all the way to zero until the chance that you'll live that long is zero.

(Of course, you could reply that your chance of living to some age doesn't go to exactly zero, but all that is necessary to prevent the mugging is that it goes down fast enough.)