Inside Star Trek: The Real Story suggests that an important difference between season three and the first two years is that the third season producers did not have the money to "junk" stories and/or scripts that were not working out (page 399). I'm not sure of the veracity of this claim; certainly, the third season junked a few potential episodes, but it may well have done so less times than in the first and second season.
Could be true. I've always heard the cause was Gene Roddenberry taking more of a backseat role, and new writers being brought on board (for less pay, no doubt). A look at the writers for each season shows quite a few new ones
(and Gene L. Coon using a Lee Cronin as a pen name).
Though before giving Roddenberry too much credit, if you notice his two third season contributions in clue such gems as "The Savage Curtain" and "Turnabout Intruder".