Just a heads up for anyone who wants to know. I was able to get Chex Quest 3 working through my steam link to my TV. Works well with an Xbox 360 controller (I don't have a steam controller)
I did switch out the Chex Quest 3.exe for the latest ZDoom.exe because I heard it had better controller support. I might try it with the current release of Chex Quest 3 though.
I can go into a little more detail if anyone wants.
Eh, it's better to use zdoom/gzdoom at this point. Chex3.exe isn't all that different and it's based on a way older zdoom version; zdoom/gzdoom have full Chex 3 support