Re: the introductory "Credits narration,"
I've got my own personal take on the Star Trek universe which may vary a bit from some of yours, but I've found it all fits together remarkably well.
First off... we know that the Federation, in Kirk's time, was "a thousand planets and spreading out." So, it's a very silly thing to suggest that there would be only a dozen "ships of the line" in that fleet, isn't it?
My own take is that there were quite a few Heavy Cruisers, with a crew of about 200, during the time prior to the "Five Year Mission." Not merely twelve. And that there were numerous other types of vessels as well.
I further view the "food slots" and so forth as limited-functionality replicators... not as "all purpose" as the ones in TNG, but the same general concept. While in TNG, we have every "replicator" being its own device, the "food slots" would be tied into some central "food replicator" system, possibly with a high-speed "turbolift food system" or a transporter-like delivery system.
In any case, I see the TOS ships... the "twelve like her in the fleet"... to be twelve heavy cruisers which were refit with replicators, and thus were able to free up a vast amount of what was previously cargo space for supporting these missions. The end result was a ship capable of longer-duration missions, with a lot of freed-up space which was converted to science labs and, well, all the things normally present only in a dedicated science vessel.
And these twelve ships were sent, en-masse, on a "five year mission of exploration" (and, obviously inferred, of expansion and growth).
The politicians would have promoted the heck out of this five-year-mission. And it's entirely possible that with the launch of each ship on it's five-year (how long is the Federation President's term? Five years, maybe?) mission, there would be a public ceremony.
SO, I see this little speech as that which Kirk gave (using bits and pieces of Zephram Cochran's "Warp 5" speech) at the Enterprise's post-explorer-refit launch ceremony.
I imagine this being played up by the politicians as a big "publicity stunt" as much as it was a "mission log." Kirk, unlike some other captains, was never publicity-shy, was he?
Why was the Enterprise one of the ships which were refit in this way, while others weren't? Well, partly luck of the draw, and partly because the Enterprise required extensive refitting and reconstruction after the "Galactic Barrier" episode, anyway.
Kirk had taken command of the Enterprise's just before the "Barrier Mission," as far as I'm concerned. He had prior command experience (while holding the rank of commander, not captain), commanding a "destroyer-type" ship of some sort (I've always treated this as the USS Alexander, a variant of the Baton Rouge class frigate hull, but not in the original configuration of the Baton Rouge).
Prior to that, he'd served on at least two other vessels... the Farragut and the Republic. (WHICH Farragut and Republic, is questionable... we don't know with certainty that either was a Constitution-type heavy cruiser, do we? All we know is that eventually there were Constitution-class vessels by those names.)
