Alternatively you can add them as a non-steam game and run them through steam

This is what I was trying to do.

I have them all set up and installed on a Games folder in either my SD card or SSD (depending on the game).

I add to Steam for the .exe and run via Proton Experimental or Hotfix or 11.

Via Desktop, the game launches, shaders cache, etc. but the screen is frozen on a gray or black or white blank screen — if I tab out or use the system key, I can see the game running fine in the background.