What makes Khan so special that the story has to be contrived to include him?
What makes Chekov so special that he should be on the Enterprise in ST09, despite his age?
The answer is that the writers are going to put things in that they think are indicative of Star Trek. Khan has become somewhat iconic, so if they could put a recognizable character in the role, they will.
The character of Chekov is a part of the TOS cast or "family" of characters. The character of Khan, isn't. Although I'll agree that Montalban's portrayal of him in TWOK made the character "somewhat iconic," at least in Trek lore. But it was still a one-off performance.
The TOS cast of characters doesn't really include a go-to arch villain like Batman's Joker or Superman's Lex Luthor. Lex Luthor is indicative of the Superman franchise. The Joker is indicative of Batman. When those franchises went to the big screen in 1978 and 1989, those were the villains each hero faced. Who else could it have been? The arch-enemies in the franchise. It's not so with Khan in TOS.
The Klingons are the arch-enemies in TOS. As a group, they are the closest thing to a Joker or Lex Luthor in TOS lore. So, as the TOS saga starts over, I guess that's why I would hope to see a Klingon as the antagonist in the second movie. That's probably why I think of Kor when I see any picture of a bearded Del Toro. Wishful thinking.
Given that in TUC we saw Kirk near the end of his career finally address his prejudice against Klingons, it would be fitting in a "reverse book-ends" kind of way to see the origins of that antagonism at the beginning of his career.