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?
Once you figure out where the save file for the game is on your PC, you can simply switch to desktop mode on the deck and copy them over. I’ve got an smb share that I used to copy things between devices, but you can also use a flash drive.