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?
Update 14/11: I now have a Steam Deck myself. After lots of trial and error, I zipped everything I found in my home folder (except the games) on my computer, then moved the archive into my Mega account. Then installed Mega on the Deck and unzipped everything there. For the games themselves, Valve seems to be thinking the same way as me - you can connect your Deck to the same network as your PC and the files to your desired game will be copied over LAN instead of being downloaded over the internet. Nifty feature.
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.