Do these sorts of techniques work in 2D animations as well? For example, certain video games will have 2D "sprites" which look very cartoonish and very much not 3D. How would they animate these types of images? Would they just draw it?

I guess we have to bound the summed transform someway so the vertex does not behave drastically?

Typically sprite animation is done with a set of pre-drawn images that represent different stages of motion and can be pieced together. This is an example of doing this in Unity: https://learn.unity.com/tutorial/introduction-to-sprite-animations