Can I run polygon insetting on the surface of a mesh?
This is more of a long comment

Yes you can do this. The problem lies in how you measure the surface. I tried to do this manually by progressively extruding tubes along the edges, and finding their intersection with the surface. It would work but is a bit hard to compute. Obviously you may want to take more steps than i did for a more accurate solution. 



Image 1 Straight skeleton (magenta)  and insets by finding the intersection of tubes along edges, and then the created edges on mesh (blue) 
From this experiment I can conclude that are many complex cases to consider.