Steam (reverse?) in-home game streaming over unRAID. Worth the trouble?

My setup: I have a PC (Linux) that I also use for gaming, but I want to connect it to a TV that's in a different room. Now, I know I could just get something like a Shield TV or Apple TV and use Moonlight and be done, but I'd have to buy that.

Since I also have a server that's basically right next to the TV, I was thinking I might be able to (a) stream steam games from my PC to my server (and then connect to the TV) and (b) send the input signals of my peripherals to my PC. I have two controllers, a Bluetooth one and one with a USB dongle. It would be nice to connect both, but that's really not a priority for a start.

Asking here because I'm new to unRAID and was hoping someone has experience with game streaming and can tell me if this would work and is worth it, or whether I should just buy a dedicated device for this. In the second case, do you have recommendations?