The only problem I see with that is during the TOS run, Enterprise encounters at least three Constitution-class ships either structurally destroyed or otherwise with the entire crew dead/lost. (Constellation, Exeter and Defiant) And this doesn't even touch what happened to the Excalibur. With those kinds of losses, such a small pool of cruisers would mean the Federation's fleet would be in a state of perpetually dire crisis. TOS shows us no evidence of that. To the contrary, it seems that the Federation has been sending out starships for years that fail to report/return to home port. (The U.S.S. Valiant, the Archon, the Horizon...)
And again... all we know is that there were twelve ships "like the Enterprise." We don't know that there are only 12 Constitution-class ships... that's not what the line was. And we certainly don't know that there aren't literally hundreds of other classes of ship in service.
Hundreds of other
classes?

How insanely huge you assume Starfleet to be? My impression of the TOS era is that there might be maybe hundred other
ships! Whit fleet of that size, twelve Connies is quite reasonable.
No matter how fast a ship may be, there are limitations on what can be done by any single ship at any given moment, aren't there?
I think you're failing to grasp just how big... just how very, very BIG... space really is.
You really need to reign in the attitude there. There's not a damned thing "insane" about assuming that there may be tens of thousands of spacecraft of various types operating within the Federation at the time of TOS. We're talking about "A thousand planets and spreading out," remember. If you assumed a single "system patrol ship" for every planet as minimal security, you'd still have over a thousand ships in the fleet.
How many cargo-carriers do you think might be in service to support "a thousand planets and spreading out?" How many tender-classes? Scout classes? Science-vessel classes? Courier classes? Passenger vessel classes?
How many different NAVAL VESSEL CONFIGURATIONS do we have, right now, today, here on Planet Earth?
How much bigger is a galactic "Federation?" How many more people... how many more tasks to be performed?
I'm absolutely comfortable with the idea that there are certainly hundreds, and quite probably THOUSANDS, of other classes of ships out there, in the same "thousand planets and spreading out" Federation as the NCC-1701 is operating around.
I, in fact, think it's "insane" to assume that there would be only a dozen heavy patrol combatants, and only a few hundred total ships, in the service of such an entity.