It might be possible but unreliable. Your best bet is to have unRAID host a VM with a VM that has either the full steam client running on it or the appimage for the steam link app. Passthrough a GPU to it (you need to do that) and passthrough eventual peripherals to it. Connect the TV to the GPU output.
For a hardware solution, I suggest finding a cheap-ish Android tv box that has an Ethernet port, and then simply installing the steam link app from the play store. You will then be able to connect your controllers to it directly, using Bluetooth or wired. It might be tempting to go to Amazon but honestly AliExpress is 60% cheaper and the same (garbage) quality. Try to find a one that says it can do 4k, that usually means it's capable of outputting 1080p.
All of this is completely disregarding the fact that you will likely have issues with game streaming when your host is on Linux. Try it out first with the steam link app on your phone.