i'd suggest that you need to consider whether what you really mean
is a "position chosen from the uniform distribution of all legal go positions",
or if you mean a position from somewhere near the middle game. (i.e. would
you be comfortable with a board with 4 stones on it as one of these uniformly
chosen boards?).
do a single playout from an empty board with a normal distribution centered
around whatever you think that the "average number of moves" is for a game
to decide when to stop playing out)[1]. this isn't a uniformly chosen board
position,
but it will very likely accomplish whatever it is that you want to accomplish.
s.
[1] alternatively, you could have a fixed probability that you "stop playing
out"
that you check/evaluate after every new move.
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