I have always thought that all the rings were "rings of rulership" that captured the purest essence and inspired the highest level of characteristic attainment for the species.
Sauron needed their participation and contribution to create the rings. Neither elves, nor dwarves, nor men would have willingly helped Sauron to make the rings for their races if they had a suspicion of his intent. That the rings exist shows they were deceived, but also that they were necessary. Sauron could have taken evil elves, dwarves, or men and given them already created rings, and with such rings captured as much of the race as he intended, perhaps more - if he had the ability to make them on his own.
Tolkien also has this idea of "noble line", shown in Aragorn. There is something about the "royal blood" that has a right to rule, and a characteristic of excellence in his books.
Given this, the Elven rings were not only the "rings of rulership" of elves, that amplified elvish capacity for greatness in their characteristic strengths, but they were something that was, in part, utterly alien to Sauron. I think that if he had them captured, that he could bend them to his will, and through them darken the hearts of all the remaining elves in middle earth.
Goblins were elves once, but they were taken by the dark powers, twisted, corrupted, broken and mutilated. Similarly Orcs were once men. This is where the Uruk-hai of Saruman come from - the half-elven analog. The Uruk-hai were the equivalent of half-elves, like Elrond and like the son of Aragorn and Arwen.
Orcs are looked at as a subhuman race. They are haters and cannibals. Even Smeagol has a soul, but it is as if the orcs in the books do not. Their wives, and children are not considered. They are treated slightly better than rats, but not as well as a good dog, and certainly not as well as horses. The greatness of their attainment is measured in things like "grond" or vast numbers in an army.
Saurons armies are strongly dominated by orcs. It may be that in the "nine for mortal men" there is a connection to the broken humanity in the orc armies. The nine rings over an orc army makes them fight longer, harder, and crueler. You can also look at the very high level of sorcery that the king of the ring-wraiths has. It seems that it put the pinnacle of human magic attainment into the arsenal of the dark powers. You can see in Barrow wights and the ghost army that necromancy is also something in the power of men - and Sauron when he re-materializes calls the nine from their graves and they come - against many bars and incantations to prevent such from occurring.
I suspect that the 3 rings in the hands of Sauron would make both the highest skills of the elves into the hands of darkness, and that it would make goblin and Uruk-hai armies substantially more powerful, unified in battle, and cruel in their strength.