The reason why they never appear in TNG, DS9 or VOY is because Rick Berman didn't like them. In fact, the main reason why is because he thought the make-up for them was stupid-looking.
IIRC, and I followed this fairly closely, Berman never commented on Tellarites.
Way back after "Journey to Babel", it was Fred Phillips who was never happy enough with his work on the Tellarites. He tried for something unusual with the deepset eyes, but the actors could hardly see, and that's why John Wheeler holds his head at a weird angle in the episode. Fred then modified the makeup appliances for "Whom Gods Destroy", and again for "The Lights of Zetar".
When it came time to develop aliens for ST:TMP, Fred created all-new aliens, but the Vulcans were back, of course, and newly-antennaed Andorians, using new, no-bake, "slip latex" techniques. The Andorians returned "because they were Fred's favorites", according to Roddenberry in interviews.
Instead of revisiting Tellarites, Fred developed the Kazarites, who were more simian than porcine. Curiously, the two Tellarites featured in ST IV are wearing ambassadorial robes originally created for TMP's Kazarites. Stock outdoor footage of distant aliens in Tellerite/Kazarite robes (from ST IV) was reused in several TNG episodes.
Berman thought that aliens with antennas were to "50s B-moive" alienish.
Yep. "Hokey" was his term. And he remained unconvinced of Andorians being believable in speaking parts until miniaturization of the electrical parts, needed to puppeteer their antennae remotely, became possible and cost-effective.
In fact, when ENT came around, he had to be convinced to use Andorians or Tellarites at all. Others were able to convince him that make-up techniques had advanced to the point that they wouldn't look "silly."
So? He was right, pretty much.
Personally, I still don't think that explains why Tellarites STILL appeared so little on ENT. They definitely got no love and I don't know why.
No one pitched Tellarite stories? Westmore was busy designing his own aliens?
You may as well ask why no TMP, ST III, ST IV, ST V or ST VI aliens appeared in TNG, DS9, VOY or ENT.
Tellarites were painted as feisty, argumentative, one-note characters in TOS. Writers tended to created their own aliens, or enjoyed developing others.
IF you asked me, I think Rick Berman just didn't know how create convincing characters very well.
Berman didn't really create
any aliens. Westmore did. Berman only had to sign off on the designs he thought would work well, or the designs that fitted descriptions in writers' scripts. In any case, Rick Berman was following a Roddenberry suggestion from the first TNG writers'bible: that space
was really, really big, and that TNG should forge ahead into new territories, not have the Enterprise buzzing around in the small neighbourhood featured during TOS. So... few Vulcans, no Andorians, no Tellarites.
That's why they probably were afraid of the rating dropping.
By not using Tellarites? Get real. The ratings dropped because ST wasn't appealing to as many members of the general public as TNG did in its first run, and TOS in syndication repeats.
