I wish I had an answer for this, Star Trek has been in my life since as far back as my memory goes, which is about my 5th birthday in 1972. But specific memories of the show are hard to come by. Most of them come from the 80's reruns when I was in my late teens, staying up at all hours to watch the show, grabbing episodes from nearby Connecticut with an antenna rotor (much like
@ZapBrannigan).
Basically I have impressions, like waiting and waiting and waiting for The Doomsday Machine to be rerun again since it was my all time favorite, and finally audio taping it. Or the time my mom grounded me for something with a "no TV for the weekend" and I, like a cocky idiot, said quickly "oh that's easy." She said "then make it a week smarty." And then I missed 5 episodes and TV Guide told me "The Man Trap" was on Friday and I tried to negotiate it down a day. "Then it wouldn't be a punishment, would it?"
Random: "The Mark Of Gideon" terrified me for years. The sudden shock cuts of the citizens pressed against the view port and on the screen. In the late 780's when all the talk of a new Trek series was floating about, I ask my mom if she wanted to watch Star Trek with me. She said "let me know if it's a new one." Starting a Connecticut rerun of "Miri" partway in and not recognizing a scene, I said "mom it's a new one!" She ran downstairs and by that time, I recognized the episode and had to apologize. Connecticut ran them uncut, the New York City area didn't, so it was new to me.
You know though, what I'm seeing here is just how scary some of us considered things about the show that people laugh at today. Things people now call "camp" were dead serious; the Gorn, the Salt Vampire, the dying Memory Alpha tech, etc. The so-called "wacky shenanigans" that modern shows try to emulate to be "just like TOS" were high concept aspects of a show that took itself seriously most of the time (Star Trek was quite intentional about its rare comedy episodes). Very few episodes come off as "wouldn't it be cool if?" that some shows, even the 90's Trek's, tended to.
Classic Star Trek lost it's power to scare me long ago, but it still carries weight and impact. The care that went into the series, the obsession over scripts which alienated established SF writers, is what makes it endure as the go to reference point of a franchise. Time passes, tastes and styles evolve. However, Star Trek remains my favorite series of all time, and not because of nostalgia. No, I never stopped watching it, so it's just part of my DNA. And my memories are too dim to associate it with the childhood feels (TAS though is a little different). It endures because it was that good.