Prodigy confirmed the Federation consists of >150 member species, not planets, so Picard indeed talks about home worlds in FC. The 1st season of PIC also established thousands of species within their sphere of influence.
Picard was speaking only of the worlds of the Federation.
Kirk talking to another Human, was speaking of all the worlds Humans were on. The majority of Human worlds are not a part of the Federation
No. Earlier in the episode, the conversation goes likes this:
COCHRANE: You speak English. Earth people?
KIRK: From the Federation.
COCHRANE: Believe me, Captain, immortality consists largely of boredom.What's it like out there in the galaxy?
KIRK: We're on a thousand planets and spreading out. We cross fantastic distances and everything's alive, Cochrane. Life everywhere. We estimate there are millions of planets with intelligent life. We haven't begun to map them. Interesting?
Kirk mentions the fed. and tells Cochrane they ("we") had barely begun to explore/ map the galaxy, obviously as being part of one unified polity (=federation). Maybe this even just pertains to UFP planets with humans. Hell, maybe Kirk does not even count all planets with some minor presence, the Vulcan P'Jem has a population of a whopping dozen monks before the spy operation gets started.
In the Lensman series space battles between the powers ruling two different galaxies involved fleets of millions of space battleships built and crewed by the millions and billions of planets they ruled.
This brings me to the subject of proper
sense of scale which, unfortunately, most writers of popular sci fi seem to lack. Realistically, considering the size of the UFP, the manufacturing capabilities, plus its population of at least two trillion in the 2250s, Starfleet should have hundreds of thousands or rather several million starships in service by the mid-23rd or 24th century. It should not be controversial, but there are people who seriously think that even the number of 7,000 vessels from Disco is excessive.
IRL, the US navy operates about 160 vessels if you only count destroyers, cruisers and submarines just to be a little conservative here. US population = 340 million Projected onto the UFP, there should be a minimum of ~ 940,000. The world's oceans are ofc a joke compared to space, while Starfleet is not only tasked with defence but also exploration, research, diplomacy etc.
In the "Legend of the Galactic Heroes" animes and novels, the total number of humans is only 40 billion, yet each of the two main factions field warships in the six-digit range. Granted it is set in a prolonged interstellar war, but still. One of the better examples of a franchise that is relatively mainstream I think