Sauron won't become the Ring Hero of Angmar, because he is already the Ring Hero of Mordor.
I agree with this, if we're looking for uniqueness (which is one of the points which makes Edain awesome). But I think that the main feature which should be the prominent one should be the lore. The lore should be more important than uniqueness to me, and I think this is also the reason why you chose to not make Elrond use the Ring.
Elrond's Ring system is different from all existing Ring systems, which a simple Sauron system wouldn't be.
Moreover, if one wanted to make the Ring system completly lore arrcurate, than Sauron wouldn't be just the Ring Hero of Mordor (and Angmar), but also a possible Ring Hero of Isengard (he even send a Nazgul to get the Ring, when the Hobbits were captured), of the Misty Mountains (they would either bring him the ring directly or be influenced enough by the ring), and even the Dwarves (Dain had the theoretical option to give the Ring to Sauron).
On the good side, Gandalf would have done nothing but giving the Ring to a Hobbit, when Theoden was healed Aragorn or Gandalf were near and would have given it to a Hobbit, Galadriel didn't take the ring herself, but given it to a Hobbit, and with Erebor Bilbo would be the Ring Hero.
The player would have Sauron with the evil factions and a Hobbit, who can become invisible with the good factions. Wouldn't that be lore wise and great?
The ring system is a "What if ..." scenario. And in the case of the Witchking it's the question, what if the Witchking of Angmar had gotten the ring, but Sauron would still be too weak to come and take it himself. So, someone needs to bring him the Ring and the Witchking wouldn't have trusted any of his servants enough to do that instead of him. But because the forces of the Witchking are under attack he can't leave imediantly and has to keep it for some time.
That's as lore wise as most other ring systems. Furhtermore, the Witchking has one of the coolest and most unique Ring systems at the moment.