And I really don't understand how this "Khan is Kirk's Joker" thing got started.
By default, I guess. The only other TOS villains who recurred -- Kor, Koloth, and Mudd -- did so only in the animated series, which has long been unfairly overlooked. And Kor and Koloth weren't played by their original actors there, while Mudd was more comic relief than a real villain.
There were attempts to make Kor into Kirk's arch-nemesis, but they fell through. "A Private Little War" was supposed to feature him, but Bob Justman thought it was too coincidental for Kirk to run into him again, given the vastness of the galaxy. And "Day of the Dove" was written for Kor and the part was offered to Colicos, and he wanted it, but he was unavailable due to a film commitment.