It's not that he wouldn't appear in the show, it's that they couldn't afford another "name" actor. Hence having Jimmy Doohan, Nichols and Barret-Roddenberry doing a bunch of the other voices.
Koenig wrote one of the episodes... "The Infinite Vulcan" I believe.
No. Koenig wasn't HIRED to be in the show, actually. But they hired him to pen an episode...
Initially, I believe that NONE of the "secondary" characters (Sulu, Uhura, Scott) were going to be on the show, for budgetary reasons. I think (if memory serves) that Leonard Nimoy protested, and wouldn't be involved unless the others were brought on board.
I believe the episode in question is "The Practical Joker". I think they simply refer to it as the "rec deck" though.
Man, I'd love to see Yesteryear, Counterclock and a few others again. (I always had an unhealthy love for "The Terratin Incident". Not one of the best.. but I always got a kick out of it.)
I'll second that, Nimoy's book relates this story that he held out unless they included the rest of the cast. But they didn't have enough money for all, so they couldn't include Keonig.
Also, the animated episode "Time Trap" was virtually copied in the Voyager episode "The Void"
Here is the Animated "Time Trap" synopsis: While exploring the Delta Triangle the Enterprise is attacked by a Klingon ship. They both disappear into a void where ships from all over the galaxy are stranded. The various races have grouped together to survive. The ships only hope is to work together with the Klingons to escape.
Here is the synopsis for Voyager's "The Void":Voyager is pulled into an empty layer of subspace where ships are forced to attack each other and steal supplies for survival. The ship's only hope for survival lies in forming alliances with other ships who wish to escape 'The Void.'