does Time Machine restore desktop items

To be clear: Do you mean that you are not finding your Desktop as in ~/Desktop and files stored therein?

If so, I suggest checking the Time Machine privacy settings. Many people treat the Desktop as a temporary storage area not worthy of backing up. The PNG file that I included below was stored on my Desktop before placing it here. eg. my privacy settings look like this:

What macOS version is the old Mac and Time Machine backup on...?

If you migrated from Time Machine during the initial setup, then everything should be as it was on the old Mac.

If you setup the new desktop first and then migrated from Time Machine, your migration went to a second user account.

Try this. Open Finder and be sure the "desktop" directory is showing. If not open Finder preferences and check to show. Then click on desktop in Finder and then scroll back in the TM drive and see if the folders show up.

They can probably find them in iCloudDrive:

Add your Desktop and Documents files to iCloud Drive - Apple Support (IE)

Normally TM will restore all to your new Mac including desktop items.