GNOME, like elementary, doesn't have Desktop icons too. But it does make the menu useful by providing only three settings there:
This falls in line with elementary's style of exposing relevant settings throughout the OS. For instance, the wingpanel indicators expose relevant settings rather conveniently. It would be great to extend this to the Desktop as well. It can also help set expectations of users coming from traditional desktop paradigms who might at first be confused as to why the right click isn't working!
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You mean making these available through a context menu / right click menu?
Yes. Exactly how GNOME has these settings exposed, like this.

I have a prototype at https://github.com/elementary/desktop, but https://github.com/elementary/gala/issues/563 will need to be fixed first.
According to people smarter than me (:wave: @tintou @davidmhewitt), the desktop context menu handling makes more sense to happen in Gala itself instead of a separate application. So I'll archive that repo and leave this open for someone to tackle within Gala itself.
This was the menu I'd built there fwiw:

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This was the menu I'd built there fwiw: