Its possible the Federation lost members between Kirk's and Picard's time due to a number of reasons.
I believe one of the prime reason the Federation was formed in the first place was the Romulan War, and the fear of the Romulans attacking in the future.
Subsequent to the formation more species would have added their star systems (or multiple star systems) to the Federation for the benefits of mutual defense.
But as time went on and the Romulans don't attack that motivation for staying in the Federation would have faded. Species would have canceled their memberships and went their own ways, by the time period just prior to TNG the Romulan War would have been nearly two centuries in the past.
There could have also been a mass exodus of members due to some beef they could have had with the Federation itself.
Some fans have posted here in the past that the Federation may have organizationally changed between TOS and TNG, going from (or attempting to) a United Nation type organization to more of a Nation State with the members require to surrendering large measures of sovereignty.
I never embraced that thought. However such a move could account for whole sections of the membership departing.
Many of them could have formed their own alliances.
Members considering canceling their membership could have said in council, "the Central Interstellar Hetaerocracy's ambassador invited our rep to diner last night, and ..."
In Metamorphosis, Kirk said "We're on a thousand planets and spreading out."
Given that Kirk was speaking to another Human, the "we're" (imho) meant Humans. Humans had a presence on a thousand worlds. There'd be all of Earth's colonies, the worlds of other Federation members and their colonies (I could imagine hundreds of thousands of Humans living on Vulcan), and worlds not aligned with the Federation of well.
We're on a thousand worlds.
I like to believe its 150+ species homeworlds, once they enter the UFP their colonies are also included--unless a colonial government opts to remain independent.
I would think that once a colony became independent, whether to join the Federation (or a different grouping) would be their decision. I could see a species homeworld not being a part of the Federation, but once independent the former colony seeking to join.
... the number 150 was chosen to parallel the United Nations.
Dominica join the UN on December 18, 1978, becoming the 150th member.
By the time of First Contact's release the UN membership had been steady at 185 members for about two years.
