I am really surprised by the large fraction of insufficient-material draws. I had expected Chess to be an unstable game even in the random limit, because the side that happens to blunder away important material first would have less opportunity to capture material, so that the imbalance would only grow. But this suggests it is actually a stable game in this limit, where imbalances tend to correct themselves.
Of course having less material reduces the number of non-captures as well as the number of captures, so that it does not reduce the probabilty to play a capture. I guess this is what I overlooked.
Interesting fundamental questions would be:
*) what happens when you bias the move choice towards captures, by a fixed factor. Would the game become unstable when the bias exceeds some threshold (manifesting itself in a huge drop of material / 50-move draws, and a large increase in checkmates / stalemates).
*) How are the wins distributed over white wins and black wins? Does white have an advantage even in the random limit, and if so, how big is it?
*) Is it a better strategy to bias towards captures (i.e. if you only bias one side to prefer captures, would it beat a random mover)? Is there an optimum for the bias factor beyond which increasing the tendency to capture becomes counter-productive (if the opponent keeps using the optimum factor)?
*) Would biasing in proportion to the victim value (i.e. play each move with probability proportional to (1+f*victimValue)) be an improvement over fixed-factor biasing?
*) Would it be an improvement to bias against moving (and capturing) with more valuable pieces?
*) Can this method be used to calculate piece values? How is the average result affected when you give Pawn odds / piece odds / Rook odds? Do you already see an effect of this in the random limit, or would it only show up after you succeeded making the game unstable by biasing the move choice in some way?
(Hint: this all seems much more interesting than playing nine billion additional random games to gain half a digit extra precision...