First, I have no use for the "Enterprise Class." Artistically, it makes the hero starship of the story too self-important, and thus bad fiction. Calling the differences a "refit." It's just a convention to prepare the audience for a bigger, prettier model of the hero ship. While it would be nice to give the completely different class would be nice, but properly, it's a refit. The name "Enterprise Class" simply comes from some dialog about a training bridge simulator that looked like a Constitution refit bridge, which would have been exactly like the bridge of the USS Enterprise. While I would chalk this up as a bit of piece lazy dialog
or flubbed lines going into production, it seems that anything that makes it to film is cannon. Even so, that could still be taken as the "Enterprise Class" only exists as a holodeck simulation. The Enterprise itself is not an "Enterprise Class" starship.
In TOS, starships were very few and very far between. I'm not talking about the ST Universe as fleshed out from many decades of the franchise, but what was presented in the late 1960's. The Constitution Class starship might have been the backbone of Starfleet, but Starfleet wasn't that big. When you watch the movies, ships were so scarce they were forced to take the new Enterprises out of stardock, before they were fully operational. In the case of TMP, STV, and Generations, they severely strained credibility. As ST advanced to TNG, there were a lot more starships. This would be natural, as the Federation grew, and more and more starships were made from an ever increasing industrial base. This would mean an exponential growth in the number of starships.
Constitution class cruisers were in service before Kirk ever commanded one. As a matter of fact, if you follow the SFB description of the development of the Fed CA, the entire engineering hull with its warp engine nacelles is an add-on.was built in under an existing sub-light "flying saucer." This makes the "Connie" appear to be a very old design. As there were fewer shipyards in the olden days, fewer ships were made, so the older the design means the more rare the design.
The Miranda class makes it debut in ST2 WoK. If one examines the model, one finds that it isn't a "light cruiser," but an advanced adaptation of the Enterprise's design with more phasers, a significantly larger torpedo bay, and twice the shuttle bays, using similar power plants in a far more efficient layout. While the descendants of Constitution design must have had some advantage allowing the 1701-A to be produced, one could easily see this design supplanting the older design. In fact, the readiness of Starfleet's to simply scrap fairly new, if battle damaged Constitution refits, shows that this design was getting long in the tooth back in Kirk's day.
In my humble opinion, the Constitutions were rare in TOS timeline, and were being produced in very limited numbers since then, with the 1701-A being the very last. By the time of TNG, most of the old Constitution Class cruisers would have been lost or scrapped. I would only expect to see one as a training ship.
That being said, I still think the Constitution Refit is the finest model ever created for ST. Every other Federation ship is a dog by comparison. The Klingon K'Tinga despite having too many gimblies is a fitting beauty. Aside from those, except a few fan-made designs, very few ST models come up to the level of "ok." I would so much appreciate if any new "Kirk and Spock" ST's would use these models. The JJ-prizes actually hurt my eyes.