I will list out those best pc games which auto plays themselves and you can leave it in livestream also. Self playing games are fun to watch and do livestream while you are doing other stuffs.

Fifa 20/Fifa 21 :

Look here you do not need to play game yourself. You can select two clubs, set time maximum that is 20 minutes (which when game starts takes around 1 hour to finish). And do not set controller to any side. Just start your livestream and do your other tasks. Remember in half time, you must press enter and resume game. 

Fifa Online 4 :

For 1vs1 mode we cannot set it to auto play. But for manger mode ranking, you can set this game to 30 matches continuously. Takes up around 4-5 hours to finish on average. During this period you do not need to even click anywhere else in game.

There are lots of other games that auto plays, but these two games mentioned are high end pc games and stream will go very friendly too. 

Game that plays itself, Self playing games pc livestream