You've nailed it--we didn't miss it. The practical effect of smaller radius is greater pressure per unit area. The smaller radius is harder on the rope. It's also harder on the tree. In two otherwise identical heavy-load setups, one with big limb and the other with a small limb, you could have a safe and normal operation with the big limb, but the small limb would get stripped of its bark and the rope surface would start melting.
My buddy Jack with 35 years in the business likes to tell me about a huge (1 1/2 in?) nylon bull rope he used to have. It was used rarely, mostly on highway projects where heavy equipment was available to pull a whole tree out of the way. On one such job the rope passed over a small crotch. The load was far smaller than the rope capacity, but Jack was shocked to find later that the rope had undergone some serious melting where it had passed over the crotch. The intense friction around the tight radius meant intense local heating, like holding a torch to the surface of the rope as it passed by. A much bigger radius would have lowered the intensity and given the rope time to absorb the heat, more like heating it with an electric blanket. Nowadays Jack always uses blocks and steel capstans to deal with big loads.
This is one of those cases where things don't scale together as one might expect. That big rope would have been almost 10 times as strong as a little 1/2 in rope, and able to do a lot more work per minute than the little rope. But it couldn't absorb heat through a square inch of surface at 10 times the rate of the little rope. Maximum heat absorption rate doesn't scale up at all, and this led to an early grave for the huge bull rope.
