As to why every single TOS crew film (I-VII) needs a Klingon reference, I don't know.
I think that, aside from Khan's "Klingon proverb" joke/misattribution, the only reference in TWOK was the Klingon ships on the viewscreen in the
Kobayashi Maru, and that was just so they could reuse TMP effects footage to save money. As for TSFS, the villains were going to be Romulans (hence the cloaked Bird of Prey and the talk of honor), but it was decided their resemblance to Vulcans would be too confusing to casual viewers, or something like that, so they fell back on Klingons.
The Klingon ambassador's scene in TVH was following up on the events of TSFS, since the three films form a continuous sequence of events. I suppose that Klaa's presence in TFF was something of a followup on that as well, and Korrd was there because all three major powers had representatives on Nimbus III.
As for TUC, it was conceived as an allegory about the recent fall of the Soviet Union, and since Klingons had always been the stand-in for the Soviets, it was only natural to use them again.
I realize that that doesn't answer the question of why the Klingons had to be reused in the second season. I don't know why that choice was made.
As I said -- Gene Coon created them, and he was the producer, the guy making a lot of the creative decisions.
However, I must say that if they'd just been a one-off, we'd be the poorer for it, if we had to lose "The Trouble with Tribbles" and "Day of the Dove".
Or we would've had other enemy-of-the-week aliens fulfilling the same role. Kinda like how
Mission: Impossible did dozens of episodes about countries that were thinly disguised USSR substitutes but they all had different names, when they were named at all.
The original outline for "Mirror, Mirror," in which the alternate timeline wasn't an evil-twin universe but just a different and less advanced one in some ways, introduced a new bad-guy race called the Tharn (a name ultimately used in the episode for the leader of a race of pacifists, ironically). They were described as having "an Oriental quality" and "perhaps finely-scaled skin" (shades of Cardassians), and are portrayed as a people bound by honor and a belief in prophecy. They might've been an interesting addition to the Trek universe, albeit even more an "Oriental" stereotype than the Klingons were.
Hmm... the Klingons never became an honor-driven race until TNG, and that was building on the "You will be remembered with honor" line from TSFS. Imagine if the Tharn had become a recurring villain race in TOS, with all the honor stuff already built into them from the start. Maybe they would've ended up filling the role that Klingons played in TNG, DS9, etc.