I'm not sure if I shed a tear, but I have to say that there's a huge difference watching the shows as a kid, pure space opera, and then watching some of them decades later as an adult with some life experience under one's belt.
The plottiness and sf elements become less important and sometimes you can pick up on issues and character development that only becomes meaningful when you have "been there done that."
At the end of "Requiem For Methusaleh," Kirk has fallen in love with the female android Rayna. Flint has brought him to the planet to wake up her emotions and it works. The android has become a human being and they are truly in love with each other. The stress of choosing between Kirk (her lover) and Flint (her creator/"father") is too much, and she dies.
At the end Kirk is back in the briefing room of the Enterprise in a complete state of unconsolable anguish. Spock shows mercy and does a mind meld and says, "Forget...." (taking away his anguished memories of Rayna).
As a kid that scene was to me a rather meaningless/silly throwaway. Oh Spock can make people forget things, too?
Only as an adult have I come to better appreciate the poignance of that scene and of the entire story.
Falling in love with a perfect android woman built by an immortal godlike man? That's actually kind of a silly plot.
But if you look at it from the inside out, as a story about a man who is dedicated to his duties, is a lady's man but has never really found a lasting love, finally finds it, then is responsible (or feels he is) for her death...it's much more meaningful to me now, as an adult, on an emotional level.
Another one that hits me the same way, probably for the same reason, is "The Paradise Syndrome." Kirk finally finds happiness and love with Miramanee on the planet, he has amnesia. But ultimately she's fatally injured and dies in his arms.
**Obviously City on the Edge of Forever is similar emotionally.