Let's say NBC decided to give Trek a chance with a 4th year, better time slot and the ousting of Frieberger, as long as Gene agreed to come back and ride herd on the series. Then bump Bob Justman to producer and bring in one more. Or maybe DC Fontana would take a promotion to Story Editor or something. This would entice Leonard Nimoy to give the show another chance. Shatner's weight would continue to fluctuate, his hair would get a little longer and Star Trek sideburns would begin to resemble muttonchops. Sulu would rock out with a handlebar moustache and the supporting cast would probably wind up with more screen time.
As the 60's became the 70's, and lighter fare gave way to relevant programming, I could see star Trek becoming darker and gritty. Along the way, Nimoy will leave the series because he would have taken the character as far as he could and would find little reason to keep plugging away. Without Nimoy, Star Trek's ratings would decline, and some local affiliates would drop NBC's early primetime lineup in favor of running Space:1999. Nimoy would be asked back for the final episode of the season, which would turn out to be the last episode of the series. It would be a decent episode, but nothing magical.
And that would be the end of Star Trek. There would be no reunions, but there would always be rumors and aborted efforts to make "a movie" with the original cast. But the fan base would be a lot smaller, as they would have had their fill with 9 seasons of Trek. With no early cancellation or the feeling of being cheated, Star Trek would be just like most long running shows of the era: Popular at the time, very well known, but not seen on TV all that much after the first couple of syndicated runs (like Mission Impossible), except on low power stations needing cheap programming.
In 1979, the world would see Lost in Space: The Motion Picture with the original cast. They would have a successful run lasting through the 80's (not having retired to Argentina, Guy Williams would not have died from his heart attack). Finally, in the early 90's, a short series of TV movies would reunite most of the cast as Captain Kirk "thunders out of retirement" to solve some space dilemma. Nimoy would not take part, and would be replaced by Wayne Rogers as Spock. The second TV movie would pit the Enterprise against Thor, the thunder god - the one seen in The Incredible Hulk Returns.
These TV movies would be scorned by fans, and show up on cheapjack video and DVD, usually found in the dollar bins at Walmart.
And in a reversal of the real 1991, Irwin Allen's death would make the front page while Roddenberry gets shoved back to page 8.