Great write-up. I get a sense that the Last Night could be described as “interactive fiction” and that makes me happy, as it’s been a while since I played games in that style.
It’s probably not two different ways, but part of a scale. I’ve seen a few games that actually combine what you write, by creating a simple mechanic-based prototype, introducing part of the setting, iterating on that, adding more of the setting and art, iterating again, etc. I think companies like Bethesda work like this. It allows you to work more parallel on both sides of the equation and the solution scales to larger teams. The advantage is that the flow goes both ways, mechanics inspire setting and art, setting and art inspire new mechanics. Disadvantage is of course, that it will be harder to make it mechanically play well and harder to make a truly immersive world.
I think each developer needs to find his place on that scale, find that spot that makes you creative and makes sense for you.