Why?
In 79 TOS episodes, only 5 of them featured Klingons.
Errand of Mercy, The Trouble of Tribbles, Elaan of Troyius, Day of the Dove, The Savage Curtain.
In 6 TOS movies, only 3 of them featured Klingons as villians -- ST III, V, and VI.
I don't know why you think that TOS means fighting the Klingons.
You forgot Fridays Child and A Private Little War. Nearly 10%. vs a mere 3 for the Romulans.
Because before TNG showed there were few races from outside the Federation that had repeat appearnaces the Klingons and their Romulan client state were the two most popular.
Don't forget that both Kor and Kang beat Kirk and Kirk was saved from Koloth by a tribble (leading to an act of ecoterrorism by Scotty). Kor had his victory overturned by the Organians not Kirk. Kang recognized the energy being as the true enemy and joined with Kirk to deny it victory, but Kang not Kirk was the one in command of the Enterprise when the enemy was defeated, the Enterprise was returned merely for diplomatic reasons.
Confrontations below the level of warfare and through client states would be a natural for the Feds and Klingons with the Organians forcing them to compete economically and dipolmatically for developing areas of the neutral zone.
By the way there were NO
Klingons in any of the movies. Kor, Kang and Koloth were
Klingons those boneheads in the movies are something else.