Now what does that mean?
It means that presenting the player with a choice to either help or not can be a very bipolar and thus hard to correctly pull off task.
An example of timed quests: Either you do it in time, or you fail the game. The primary matter involves no decision at all; the task itself might, but you have to manage it in time in order to play further. The player is pressurized, the devil on your back will stay until you relieve yourself of this burden.
Another example that goes for any quest: Either you do it, or you're not a hero. It's basically not a choice (the archetype of 'I play a hero' or 'I don't play a hero' is set) and thus plays out like the first example - you fail, you reload.
Then there's: Either you do it or you betray a certain faction/help another. There's no choice in this particular matter - you have decide beforehand which faction you'll support. At least the time limit will not allow you to rethink faction allegiance if you've not yet made this choice.
These are streamlined experiences, but much like the character you create, there should be room for ambiguity and thus spontaneity. The decisions you make shouldn't be set in stone because you're only allowed to play a number of archetypical characters. Timed quests should really ask of us the question if we want to involve ourselves in the matter or not. There should be fun in not choosing to do anything.
Of course you have choices if you decide to do something or even if you don't. But this simply relocates the choices to another stage.
Now how can you shake up the formula a bit? One simple thing is failure. Known or unknown risks. Surprises.
You're no Witcher, no superhero and no Dragonborn. Let things happen beyond your control. Let certain affairs be beyond your abilities as a human. This is something that should go for timed and non timed quests alike. Example: You CAN'T save this town. But you CAN try. The result of this is depending on how you try, but it will be to your displeasure.
Another way is moral ambiguity and factional nonrelevance / relevance for only one faction.
You should combine all of the above. The 'do nothing and fail the game' one is optional for me.