I don't know a way to download multiple file at the same time with iCloud Drive.
But you can maybe solve the problem with AirDrop. If you have multiple files, for example in Finder on your Mac selected, you can send them all with only one click to your iPhone. The same works also from iOS-Device to iOS-Device.
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
After several comments and clarifications, the iCloud sync process did eventually start. It took about 12 hours of leaving the laptop plugged-in and awake for the syncing to start.
Because there's no UI (this is supposed to be a background process), you really have no good way to measure the progress. Unfortunately, this has been a knock against iCloud for many years and Apple doesn't appear to be making any changes to improve in that area.