I think no one will dispute this as a classic episode. It's the classic episode, frequently cited as the best episode of the whole franchise, not just original series. That makes it essential viewing, yes?
Simply put, "City on the Edge of Forever" legitimized the whole Star Trek concept. It was the first time the series managed to completely transcend itself, produce a story that could be viewed and appreciated by anyone. Its science fiction was sound. Its social commentary was nuanced. And it even managed to handle the main characters well. Is there really anymore to ask from an episode? It's the epitome of Gene Roddenberry's idealistic vision.
Somewhat perversely, that vision is explored by a guest character whose rigid definition of improving the world around her would have led to apocalypse, had it been allowed to play out. The flipside of every time Kirk interferes with an alien culture, the very argument for the Prime Directive, is the idea that the right thing to do...isn't always the right thing to do. It's...fascinating, as Spock might say. "City" won a Hugo, sci-fi's Oscar. That's how much of an immediate impact it made, just the kind of reception the series needed. Might it be said, in fact, that "City" is responsible for the fan response that has kept it going for half a century?
Such is the love for the episode that Harlan Ellison's original version of the script is as equally treasured as the final product. That just doesn't happen. That's a matter of fans dedicating themselves to a single episode the way they would to a specific series in later years.
The emotional crux of "City" revolves around Kirk in his most famous romantic relationship. He falls in love with the social worker who is prevented from dying by McCoy's trip through the Guardian of Forever's time portal. It's one of those stories where history is changed but those who were there during the circumstances where it changed remember how things originally were. This leads Kirk and Spock to go back in time in pursuit of McCoy, but they arrive before he does. Half the episode is just following these developments, waiting to see how fate plays out. When the moment arrives, Kirk realizes he has to let the woman die. Perfect moment.
If you watch, or recommend, only one episode of the series, this one is it. And it's a classic that ages better than any other episode of the series.
franchise * series * essential * character