Forget 2 long rests as a time limit. That is not what Im saying. 2 long rests was an example. You are focusing on the specifics, not the concept, My proposal is that you would still have lots of long rests before something truly bad would happen.
What Im actually saying is:
Complete the quest in x days, get the best reward
Complete the quest in y days, get not as good reward and trigger a cutscene to explain why event has not yet happened
Complete the quest in z days, get an okay reward and trigger another cutscene buying you more time
Then, after much time goes by, if you still ain't done the quest, something bad happens.
That was the idea.