Very interesting thread. Let's take a critical look at the turbolift scene in "Tomorrow Is Yesterday":
KIRK: Bridge.
CHRISTOPHER: Must have taken quite a lot to build a ship like this.
KIRK: There are only twelve like it in the fleet.
CHRISTOPHER: I see. Did the Navy..?
KIRK: We're a combined service, Captain. Our authority is the United Earth Space Probe Agency.
CHRISTOPHER: United Earth?
KIRK: This is very difficult to explain. We're from your future. A time warp placed us here. It was an accident.
CHRISTOPHER: You seem to have a lot of them. However, I can't deny the fact that you're here. With this ship.
Note that the conversation between Kirk and Christopher seems to emphasize the parent organization and loosely implied industrial/military/financial might to build such an impressive spacecraft. At no time during the conversation is the Constitution-class of Federation starships, or even the notion of starship classes, mentioned or implied. It is therefore reasonable to observe that Kirk's "only twelve like it in the fleet" remark is not necessarily a meditation on starship class or classification at all.
Kirk appears to be talking to Christopher about (1: identifying the Enterprise and her crew as friendly and non-threatening, (2: generally of Terran origin, "little green men" or Vulcans aside, (3: that the "authority" responsible for producing the impressive spaceship are the future "United Earth", specifically a "combined service" organization called the "United Earth Space Probe Agency". In this specifically articulated context, it should be clear that Kirk was trying to convey that the Enterprise and her crew are from Earth,
not that he was trying to educate a 20th century U.S. Air Force pilot on the specifics of starship classifications.
So, was Kirk's "twelve like it in the fleet" a line about the specific number of Constitution-class space vessels, or something else? It isn't really that clear. It could have been 12 Constitution-class vessels from Earth, or it could have been 12 U.E.S.P.A.-sponsored starships-of-the-line as part of a larger Federation fleet, or something else entirely, like maybe only 12 starships (out of a pool of a much larger number) are outfitted for 5-year missions of deep space exploration. The thrust of the conversation is about trying to assure Christopher that "you're among friends", not to school him on fleet strength or technical details.