Having a variable time limit is a good thing in my mind. It forces the player to accept that can't do everything in a life time, and the goal becomes how well you can do in your chosen life path. If the game offers a fair amount of diversity then it all increases the reparability as the player tries for different and better endings.

Did they marry the most beautiful woman in the city?
Raise a healthy family?
Pay off the debt?
Recover the Family Estate?
Earn a large fortune despite their crippling challenges?
Lead the city to victory when the raiders tried to sack the city?
Become the most famous trader?

All of these can make for a fun experience as the player tries to do as much as can with the time they have.

That first one... that is incredibly stupid imo. Unless there is more to it, but imagine just playing a character in lets say.... WoW. a 40 man raid, and 2 of your guys die because.... just because (according to how you're wording it) It would make no sense, they aren't attacked, and away from danger and yet they still died. Of course, it doesn't matter much in WoW, since you can revive but if it's a hardcore character that has a one time death limit, it would just destroy any want to play the game.

If you were to apply health status to it, it wouldn't be so bad. Example:

a 1 to 100 meter bar, measuring your health.
At 100 you are at perfect health, physically fit, etc.
At 50, you are susceptible to illness, like... Aids, The Common Cold, the Flu.
At 0, you are very likely to be terminal.

With something like this and a way to raise your health with actions, it would be more reasonable and not catch the player off guard and ruin it. If they do decide to not care about their health, then they really have nothing to complain about.