Though I think if you wanted to do a series, you'd probably want a little more conflict than a paradise planet could offer.

I could see a series in the vein of "The Love Boat" centered around Risa.

That's difficult, honestly. Most of the novels I read don't put much of a focus on the societies they live in, more on the characters or cool phenomena. I personally liked the depiction of Earth in the Department of Temporal Investigation novel series (which, by the way, is excellent anyway), but even that wasn't very specific.

As for fan fiction, I personally try to write much more plausible fiction that doesn't take "human-ish" patterns for granted; e. g. some of the species we explore don't even form nation states, I put more of a focus on non-humanoids, I try to make Starfleet and Federation names and representation equally distributed among member species (e. g. no USS Einstein, but instead like USS Rogra jav Baur, after the Tellarite diplomat), and look at super underrepresented peoples, subcultures, professions and areas to flesh them out a bit. I also assume that in the future we are talking about, important places are all over Earth, not just in the USA and Europe. Like, the hero ship freighter that I am writing about currently is called the SS Kyakhta, after the Russia-China trade route in the late middle ages. The Captain is a non-binary elderly Kaferian. And the only human crewmate is from Daşoguz, Turkmenistan; which developed into quite a bustling center for high-quality engineering schools.

But I haven't read much other fan fiction with the same values; most Trek fan fiction is centered around the main characters of the shows, usually in a romantic or sexual manner. Not that that's bad, I just wish there was more general, plausible-for-a-show fiction too.

Yeah it was a post-credits scene