Are open world elements detrimental to story telling? I don't think so, all that's necessary is for the side-quests and fucking around stuff to be tied into the main theme/narrative.
A good example is Saints Row The Third, you can spend hours and hours dicking around outside the main quest in that game, but it still feels like you're progressing the narrative because your actions give you control over Steelport, which is the aim of the main quest anyway.
Skyrim is a different example, where there isn't really a main quest (technically there is, but it's not central to the game which is what I mean), it's much more a game about making your own story, becoming the badass hero and whatnot, and therefore all the sidequest stuff is perfectly relevant because it serves that purpose.
Far Cry 2 is also a good example, it's a game with deliberately ambiguous goals, so every action feels relevant simply because "I dunno, it might work, might not, same as the "main quest".
Yes, sometimes the main storyline feels stop-start, or pointless because of the open world elements, but I don't think that's an inherent problem.
However, your idea isn't totally without merit, time limits on missions can be a good way of providing engagement as well as consequences for actions. In Deus Ex HR (spoiler I guess? It's right at the beginning) at the start if you dick around Sarif HQ too long at the start the hostages get shot, which makes it feel like the story is reacting to your actions.