Seems ol' Gene Rodenberry had his finger on the answer.
A culture that has mastered faster than light space travel, will have also developed faster than light communication technology.
I agree. He was unusually imaginative, and had an eye for the big picture. Also (and as you probably know) he recruited some of the most talented writers in the SF world to write some episodes of TOS, which I think surely deserves a place in television's "Golden Age."
The episodes The Cloud Minders and A Taste of Armageddon were among those which stuck with me for the rest of my life, and are relevant today.
Also the episode And The Children Shall Lead was (I think) Rodenberry's gentle criticism of the youth/hippy movement of the 1960s. If you take it in that way, it seems to indicate that Rodenberry was fundamentally a conservative.
Also, it could be argued that he was possibly the first producer of a popular program to introduce what we call today multiculturalism, by including human characters from many ethnicities. Of course there had been Black characters in leading roles on TV before (Bill Cosby in I Spy comes to mind), but Lt Uhura was not only Black but also African, as you may recall from the episode in which had her memory wiped out and had to be taught everything from scratch, starting with Swahili, her native language. I believe that was the first time I ever even heard the word "Swahili," and I'm sure that was the case for many other eleven-year-olds.
The whole "subspace communication" thing illustrates that Rodenberry was aware of exactly what you stated, that the speed of light is much to slow for a civilization that can truly explore the universe.