iCloud does not backup all of your data. It's more a replacement of iWork.com sharing in previous (or current?) versions of iWork. It only works with applications that specifically support iCloud:
If you have the same iCloud-enabled apps on more than one device, iCloud automatically keeps your documents up to date across all your devices. So you don’t have to.
If you visit http://icloud.com and click the ?
link, you'll be sent to the help site, which states:
Documents in the Cloud overview
iCloud keeps your documents up to date across all your iOS devices and on icloud.com. Every time you edit a Keynote, Pages, or Numbers document on your iOS device, your changes are automatically available on your other iOS devices set up with iCloud and on the web at icloud.com.
This also explains why the iCloud site does not show a generic documents section in the overview, only iWork. Of course, developers are free to support iCloud in their own applications.
The supported file types are listed here:
You can also drag and drop a new iWork ‘09 (Keynote, Pages, and Numbers) or Microsoft Office (PowerPoint, Word, and Excel) document from your computer to icloud.com, and then access the document from Keynote, Pages, or Numbers on your iOS device set up with iCloud. Supported file types include Keynote ‘09, Pages ‘09, Numbers ‘09, PowerPoint, Word, Excel, .txt, and .csv.
Doesn't look like there's a way to single out the Desktop via the 'Manage Storage' GUI.
However, you can always just symlink any directory from your iCloud drive; that way you can ensure that only the Desktop folder is being synced. This is with the caveat that while your Desktop files are being stored in iCloud, they're still taking up space on your HD before, during, and after sync. If you do it on every Mac then you'll keep your Desktop in sync between them all (files added, deleted, changed etc); this might not be what you're after though.
If symlinking isn't something you know how to do, instructions are below.
Backup (via TimeMachine or whatever you use).
Then fire up Terminal and run the following to move your Desktop folder into your iCloud Drive:
mv ~/Desktop /Users/<yourusername>/Library/Mobile\ Documents/com~apple~CloudDocs/
Then just symlink it to the usual 'Desktop' location:
ln -s /Users/<yourusername>/Library/Mobile\ Documents/com~apple~CloudDocs/Desktop /Users/<yourusername>/Desktop
Now anything that you add to ~/Desktop (by dropping things onto your desktop etc) will actually be dropped into your iCloud drive. They are of course still taking up local space on your machine. When you delete something from your desktop then it's gone from the 'Cloud' as well. The benefits of this are more for sync between machines (+ access via your iPhone of course).
If you've got another mac, move anything you value from it's desktop folder into your iCloud Drive 'Desktop' folder first, then delete ~/Desktop on that machine and symlink again as above.
Best Answer
Apple has a built-in setting to do basically exactly what you want. It's not technically a symlink, but it accomplishes the same functionality. It also gracefully handles different desktops on different machines. And it all works as expected in the Files app on iOS:
https://support.apple.com/en-us/HT206985