If the grass in the fins (not the mud) was black, it's been way too hot. You did say there was a broken ring, but on the rod that didn't break. I can't see any way the crankpin could have got into that condition if oil had been flowing across it. Overall, I'm not comfortable with the theory that the problem was caused by overheating alone, or even primarily.
I'm guessing you are leaning toward replacing the engine, at those prices. It gets you back out there mowing for $1,000 with a new engine. Life could be worse.
