The original Thief was more about the surfaces you were walking on and how much you were moving, mostly independent of your stance.
I don't have a good answer for what the catalyst was, but I imagine it serves as convenient shorthand. It makes more sense in a non-dedicated stealth game to have a button there to toggle between modes.
I feel that it's worth saying that if you can't get a handle on the esoteric gameplay of Metal Gear (which I would completely understand), or really any series, you don't need to try to convince yourself that you should like it because its reputation suggests so. It's kind of maddening to not see what you assume should be there, but at this rate I don't think you're going to suddenly turn a corner on anything - outside maybe the modernization aspect of the fifth game.
There's not exactly a lot of the Metal Gear narrative that I'm going to absolutely stand by, and I just don't think "playing for the story" in spite of everything else is likely to be all that rewarding.
I mean, as I understand it the aesthetic you're describing is derived from flat shaded polygons that aren't textured - and that's an early days of 3D thing. I think modern graphical techniques on top of that look better than anyone was really ready to give credit to. And also realize that we're not that far removed from "low-poly + bad textures" as a technical limitation, like a Peace Walker, so I think the impetus has probably been on moving away from anything like that for a long time, aside from some indie experiments.
But I bet some of the aesthetic's revival comes from Journey. It's not really the same thing, but it did get a lot of eyes on a sort of minimalism that shares some common features, but also it's not a super tightly categorized look anyway.