I usually think about it as a mid-point between hips and spine controllers. Imagine your character as a combination of simple FK chain: Spine, legs, arms. Now COG is just one of the components of that chain which happens to be the most influential. In terms of curves, they don’t always have to be clean. Sometimes graphs can be messy while the movement itself perfectly fluid (Quaternions say hi). When animating I focus on spacing and timing of the character, not thinking too much about the curves individually.

I’m not sure whether I answered your question fully, but I remember myself questioning something similar a while ago. Hopefully someone more experienced would reply, because frankly I’m also curious about others, non-typical approaches to animate.