[The Squatter of Amon Rûdh]

Got it.

If Sauron had made several ruling rings, they'd have contained so much of his power that without them he wouldn't be able to stand up to an elderly gnat with a bladder complaint.
The One Ring is probably like many things in ME, such as the Two Trees and the Silmarils: once made, never to be replicated.

Plus there's a question of job division: If we have several Ruling Rings, it begs the question: wouldn't they each want to rule the others?
Do Rings bicker? If so, wouldn't Sauron end up like an harassed mother surrounded by squabbling children? Viz: "Look, if you buggers don't stop it this instant, we're all going to take a little walk to the Cracks of Doom!"

Just a thought.