According to "Samaritan Snare":
PICARD: Well, I was a young Starfleet officer, not many years older than you are now. Top of my Academy Class. Green as hell. And oh, so cocky. Too cocky, as it turned out.
WESLEY: What happened?
PICARD: Several friends and I were on leave at Farspace Starbase Earhart. It was little more than a galactic outpost in those days.
WESLEY: Was this before the Klingons joined the Federation?
PICARD: That's right.
That is the end of the discussion. The statement that "the Klingons joined the Federation" during Picard's career in Stafleet is clearly established to be a true statement by this dialog. There are many other important parts of Star Trek lore established by only one or two sentences of dialog in a single episode.
It must be true - from a certain point of view of course. It might a correct but incomplete statement.
Imagine a secret alien observer recently sent to Earth talking to another one who has been here for generations. Suppose that the older alien observer mentions some long ago event and the younger one asks if that was before the Humans joined the United Nations and the older one says yes.
It is certainly correct that (some of) The Humans joined the United Nations. It fact the vast majority of Humans are citizens of nations that joined the United Nations in over a hundred separate acts of joining the United Nations.
Or suppose the alien observers dated some event relative to the time that the Roman Empire conquered or annexed the Roman Empire. That would be correct. There have been times with more than one Roman Empire simultaneously. Such a statement would be a bit confusing since there has been more than one time that a Roman Empire annexed another Roman Empire.
Or suppose that the alien observers dated an event relative to the time that the Boer Republic was annexed by the British. Wikipedia lists 17 Boer republics and sometimes there were several different Boer Republics at the same time. Britain annexed one of them, the Natalia Republic, in 1843, and another one, the Transvaal or South African Republic, in 1877.
The Boer ethnic group and the various ethnic groups of their slaves and servants can be compared to the Klingon species and the various species of their subjects. So the various sometimes contemporary Boer republics can be compared to the various sometimes contemporary Klingon empires that may have existed during the course of Klingon history.
There is no more reason to assume that there must be one and only one Klingon Empire than there is to assume that there must have been one and only one Roman Empire or one and only one Boer Republic.
And in fact in "Sins of the father":
PICARD: We're changing course. Set coordinates for the First City of the Klingon Imperial Empire.
This implies that there are so many Klingon empires and other realms that Picard has to use the full official name of the Klingon Imperial Empire in order to be clear which Klingon empire, planet, and city he means.
It seems clear that the Klingon Imperial Empire that K'mpec is the chancellor of in "Sins of the father" is the main successor state to the Klingon state that Gorkon was the chancellor of in Star Trek VI: The Undiscovered Country. And the Klingon Imperial Empire is probably much larger than all the other Klingon states combined. For example, the Klingon Imperial Empire might rule a thousand star systems while fifty other Klingon states only rule a single star system each while ten other Klingon states rule ten star systems each for a total of 150 star systems ruled by the minor Klingon empires and other Klingon states.
And clearly at least one of those Klingon states has joined the Federation as a full member or associate member. If that Klingon state kept its military separate from Starfleet as a sort of analogy to a state national guard in the USA, Worf could be the first Klingon in Starfleet.
So there is no logical reason to assume that the later Klingon story arc invalidated the statement that (some of) "the Klingons joined the Federation".
The dancing fingers type, and having typed,
move on: not all your genius nor wit,
has power to retcon half a line,
nor all your tears erase a single word.
Did the Klingons join the Federation?