Star Trek: The Motion Picture? Eye of the Beholder? I think one of the issues of the comic book "Crew" (featuring the shakedown cruise of the original Enterprise) might satisfy your story wish.
What do you mean? We saw Malurians having transporters in like the tenth episode, and it wasn't treated as a surprise or a shock or anything. The only thing I can remember towards that end was them saying that Andorians didn't have transporters yet in "The Andorian Incident". I mean, I'd agree that they implied that it was something where humans were ahead of Vulcans, but I don't remember ever feeling an implication of it being something where humans were ahead of everyone.
I'd like to see this too, if it were a political story. I think it would be cool if they were debating a new, revolutionary design and the engineers want one thing and Starfleet admirals want another and there's a bunch of potential captains vying for command. There could be fights about shield tech or weapons power (how much is too much?) and one guy says they should have enough room for families and then everyone loses their minds. That could be fun. It would break Gene's "no conflicts within Starfleet" rule, but we all gave up on that a long time ago, right?
I recall one of the Lost Years novels featuring Chekov or Kirk or someone running around the drydock and incomplete Enterprise after somebody.
The background story of The Tech That Made Bathrooms Obsolete (whether that would be transporters or something else)...
We saw the first test flight, but not the development and construction. I think there's a good story there.
^Something along the lines of a Hidden Figures story? Little known person who invented/did something crucial to Cochrane's work but was all but lost in the pages of history?