I agree with you in that, only a time limit for some quests. If you had a time limit like the one in the other fallouts would not be very good. A lot people don't even though there is time in F3 and FNV
But what is the difference though? In my example the dying man had a week to live (say), and in Fallout 1 the Vault 13 denizens had 21 weeks to live (with option for more if you sent them water*).
The one difference that I see is that their deaths end the game instead of just ending badly [for them], and the PC moves on; But their life (V13) was the reason you left, fail that and you fail everything.
The Waterchip quest was a "boot you out into the world" starter quest that kept you running and the PC realistically could not afford the time to leisurely explore unrelated tangents. If you were sent to call an ambulance, would you stop to help someone get their cat out of a tree or carry their groceries? (once the emergency is handled, you could go back). Fallout begins with a world where you can go anywhere you like; but starts you out with a "Life or Death" pressing concern that must be attended to ~FIRST. How else (or better) to impress this upon the player than to end the game if they fail.
Imo, time trigger (as opposed to event trigger) is bad for an RPG....unless it is focusing on time (i.e. time traveling) and have slower game time ratio.
Myself I don't like event triggers, (unless very carefully handled). In FO:NV I walked the PC though an area that was lifeless and abandoned, down the road I accepted a quest and it had me return to the very spot to eliminate a threat that was no where to be found (10 minutes before), but was suddenly crawling with enemies.

:thumbsdown:
* I was disappointed that they were not there to begin with, and that I was not given the "Been there Done that; Your welcome" option in dialog with the quest giver, had I encountered their threat and dealt with them already. Same thing happens again when they ask you to visit Nipton and you've already been through there..