I have an exclusive gaming PC. Ideally what I'd like to do is shove it in a closet or other vacant room to contain the heat and then stream games over LAN.

The problem with existing solutions is:

1. They need a monitor connected
2. The monitor to be powered on
3. They don't support varying refresh rates/resolutions. Only whatever is displayed on the connected monitor. I want to play on my 4k/60Hz TV in the living room for AAA visual spectacle games that work best with controller, and the 1440p/120Hz display in my office for FPS or otherwise fast-moving games that demand KBM, and the 5k monitor for photo/video editing.
4. I haven't had much success getting these working at all. Red screens, 1/4 screens, image noise, etc.

They're literally just mirroring the screen of the connected display.

I really don't want to have to buy a dedicated PC for each use-case. Does a solution like this exist? What are their pros and cons? Preferably something that doesn't require a degree in software engineering.

Currently I am using Bazzite OS with 5700x + 6800xt, if that matters.