I think it is important to keep in mind that the focus of the respective scene
is reverse causality, prophecy, people from the future telling the guy from the past what he has to do in order to create their very future or however you wanna call it.
It's kinda like with Max Weber's ideas about Calvinistic predestination being the cultural foundation of capitalism. Like in the case of Cochrane the natural question is, if everything is already determined why not just be lazy and enjoy your life? And I think the answer to this question which Cochrane found when he decided to do the warp flight the next day says more about what Star Trek is about than the uttering of the words.
So yeah, I'd say that the scene works not because of but in spite of the words Star Trek being uttered. The words are just a clue, they just indicate that there is something quintessentially Trekish going on here without feeling pathetic as "astronauts on some kind of Star Trek" has, if anything, a slightly condescending vibe.