I was in your position. I have to say the ideal solution has been: I just put Bazzite on my PC and moved it to the lounge as a console PC. No streaming required.
If you're a first person shooter pro then you might want the PC at the desk for KB+M control. Otherwise for other games (like strategy games) it is better to stream from lounge to PC for those few games and have the native PC experience with the TV.
I mostly use my desk PC for work, and any cheap old PC can do this. I got a basic mini PC workstation pretty cheap, and this OS actually good enough to play most indie and emulated games natively at 144 FPS anyway. Something even older and cheaper would be perfectly fine for my workstation needs.

