Interestingly, Skyrim was exactly the thing I thought of when thinking of this trope's origin, so there could indeed be something to it.

Recently however, I've seen an interesting spin on it in The Blood of Dawnwalker. There, too, you are faced with a time-pressing issue as part of your main quest, but the difference is: it is actually time-pressing. You can fail one of the main objectives of the game by not being there in time. And while it is debatable whether it's good for the exploration aspect of the game and whether it presses the player too much, at least it is consistent.