How clean should the outside of pots and pans be scrubbed? I’m the official pot-scrubber in our household. I have a question on how clean one should scrub the outside of stainless and copper pots and pans and whether this is merely an aesthetic choice. 

I note that cast iron cookware shouldn’t be scrubbed at all. 

Does keeping the outsides of stainless and copper pots and pans serve any purpose other than aesthetics? For example: might heat transfer be better if the bottoms were shiny instead of blackened?
I think this is purely aesthetic. The outside of my stainless cookware looks terrible. The oils polymerize and are really difficult to clean after that point.  I don't worry about it too much, though I have a friend who takes great care to keep his stainless shiny.  I doubt the buildup is enough to impact heat transfer.