Another obstacle is that whereas an image classifier is basically a purely learned function, Monte Carlo tree search has a hard-coded element of applying potential moves to find new states to explore - always following the game’s rules correctly. If you go more than a few moves ahead, it has a relatively unpredictable choice of what states to explore; but as you get closer to the current move, I think it starts to look a bit more like brute force. So if you want the AI to make a really bad move - one that has severe consequences in the next few turns - you need to not just make the NN think it looks good, but that the game state still looks good even after you make any possible counterplay.
Disclaimer: I don’t actually know much about either neural networks or Go.
(On a side note, the more I think about Anthony et al 2017/AlphaGo Zero's tree/expert iteration, the more beautifully simple it appears to me.)
AlphaGo Zero has played itself, and they have published 20 of those games.
GAN's were not used for alphaGo, as the article points out Deepmind uses reinforcement learning and MCTS.