I don't know a lot about this Go example, but using a uniform distribution for the evaluation function sounds surprisingly similar to another phenomenon I observed.

Suppose you have a linear evaluation function - h(x) is the value of something. Suppose also the coefficients of h are all positive. Then you'll be right 75% of the time (averaged over all possible h, drawn uniformly from the unit simplex) if you just approximate h=[h1,h2,...] by u=[1,1,...,1].

http://www.chrisstucchio.com/blog/2014/equal_weights.html

So I agree with this claim - uniform distributions are fairly robust to errors. But I don't think that's particularly related to randomness - Monte Carlo is only needed to integrate the distribution.

It's also worth noting that adversarial situations (like Go or Chess) are considerably different than most other cases. In a true adversarial problem, there is no probability distribution - the opponent is omnipotent. The purpose of randomness is simply to reduce the power of the adversary's intelligence - in a completely random world, intelligence is useless.