I did a bit of research in the field (You know I'm something of a scientist myself). I wrote a little Python game (unplayable, just to see the AI's fighting) which uses genetic algorithms and plays a looot of fights to try to find the best weights to outperform the opponent.
I'll leave the link Here in case someone wants to see the code.
So, yes! You can use genetic algorithms to train neural networks to play against humans on very simple games. I believe this approach could be used on more complex games, and even combined with traditional if statements algorithms.
To finish, huge thanks to the authors for its article as huge in quality as in size.