I have to mention that my computer is also running Linux, more precisely MX Linux. And yes, I know that Steam Cloud does exist. My concern is with games that do not support it (or not for certain things, maybe). Would manually moving games from my PC to the deck work? Maybe through mounting the Steam folder on the deck as a network drive or some sort?
Specifically what are you asking? If your pc is powered on and Steam is logged in, and your Deck is on the same network, you can click install on your Deck and the files will transfer over your LAN.