I think I experienced the same thing yesterday while playing farming simulator. In the middle of the game the screen went black and and afterwards my Chromecast rebooted.
The game session was not recovered (usually when inactive I have 15 minutes to rejoin the running instance).
Reading your post makes me think farming simulator is bugged specifically. I have not seen a black screen of death with other games.
Bit off topic, first week I could join voice chat, but since a couple of days I get errors saying I need to connect a microphone. Doesn't it work through the built-in assistant microphone? Is this a bug, or was it a bug that is now fixed?
You can't, and have never been able to this far, chat using the Stadia controller unless you have a 3.5mm headset connected to it (soon, hopefully, also USB headsets).
It does sound like a bug related to Farming Simulator though as I've been able to switch to headset play in other games without issue.
Best bet is if it really starts to bother you is reach out to the Farming Simulator developers. https://www.farming-simulator.com/support.php?lang=en&country=us
it's not a farming simulator issue it does this on every game you cannot play any game on Chromecast with voice without it crashing Chromecast and it instantly does it for me as soon as I plug the headset in it starts stuttering and skipping until the game finally locks up and crashes the Chromecast or disconnects the controller and won't let it connect until a restart is performed