In my experience, virtually all of Calc III is "visualizable" (even somewhat esoteric stuff like Lagrange multipliers[1]), because it's mostly about vectors (which have a natural geometric interpretation). My claim would be that to be really good at math, you need to be skilled at both visualization (and other intuitions) and abstract systems. They complement each other well.

[1] E.g. http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.ht...