Filmation only produced 22 episodes of TAS--that's barely a standard single season of the period, so to complete a televised 5 year mission, you would need at least another 22 or so TAS episodes.
Remember, at the time, Filmation's practice (and what was ordered by networks) was to produce up to 16 episodes in a season, and if renewed, produce another 6 or so, with the network padding the second season with reruns from the first. You will see the same kind of truncated second season not only with TAS, but Filmation's The Brady Kids (ABC, 1972-73), and The New Adventures of Gilligan (ABC, 1974-75). The point being that in counting a certain number of episodes from TOS making up at least three years of the 5-year mission, to complete it, ST would need more than an additional 22 episodes.
In some TOS fan circles, they believe TOS was the complete 5-year mission, since they consider the significant overhaul between "Where No Man Has Gone Before" and "The Corbomite Maneuver" demanded a long period of time, thus "unseen" missions adding up to two years took place in that period.