is there a way to only sync files which are on my desktop?
I don't want to sync my whole
~/Documents/ directory because it's very big.
Yes and no. The no is Apple is all or none with iCloud sync of files - you get documents and desktop sync when you enable iCloud sync on macOS.
But you can make a new folder called local Documents or whatever and move all the files you want locally to the new folder, then enable cloud sync. Your Documents proper folder is in the cloud as is the desktop, but the Documents will be empty other than what you intentionally want to sync up.