A few minor nit-picks to consider; feel free to correct me if I'm mistaken...
No character in TOS or TAS ever calls the Enterprise or any other ship the "Constitution class". This reference was used on Scotty's monitor in "The Trouble with Tribbles", but never spoken. It was no documented until Franz Joseph Schnaubelt's Technical Manual and starship blueprints were published in the mid-1970's. The only time I remember a character referring to the Enterprise as a Connie was when Picard read a historical computer readout of the TOS Enterprise in "The Naked Now". That would've been 1987.
In teaser/hook of "The Doomsday Machine", Spock makes an interesting reference once the Enterprise detects another space vessel. Consider this exchange:
KIRK: Every solar system in this sector blasted to rubble and still no sign of the Constellation! Matt Decker's in command. What could have happened to him?
SPOCK: Captain, the two inner most planets of this system appear to be intact.
PALMER: Sir, I'm picking up a ship's disaster beacon!
KIRK: Try to raise it, Lieutenant!
SPOCK: I have it on the sensors, Captain. By configuration, a starship! Stopped in space. She appears to be drifting.
PALMER: No answer, Captain. All I get is the automatic beacon.
SPOCK: Sensors not recording normal energy output.
KIRK: Approach course, Mister Sulu!
[We see the Starship Enterprise flying through space. Then we see another, familiar-looking starship on the Bridge's veiwscreen.]
KIRK: It's the Constellation. Look at that!
Note that Spock simply identifies the just-sighted vessel as "by configuration, a starship". No class nomenclature or anything. This seems to reinforce that, at the time of TOS being in production, everyone was still imagining that Earth / the Federation simply deployed a "star ship class" of ships familial with the Enterprise. (That is, after all, what the dedication plaque on the Bridge set read.)
In "The Naked Now", Picard appears to read from the record a reference to "U.S.S. Enterprise, Constitution class", but the data on the display appears to show a TMP refit configuration, not the TOS appearance from "The Naked Time". This seems to suggest that, in the eyes of condensed 24th century history, all refits of the Enterprise, from "The Cage" configuration to the "Where No Man Has Gone Before" update, to the subtle upgrades throughout TOS, to the various refits, upgrades and so forth in the TMP era, are all considered Constitution-class starships. This loosely suggests that the Connie nomenclature is an overall indicator of a basic design, not its constantly evolving particulars.
If one subscribes to Franz Joseph Schnaubelt's Technical Manual and starship blueprints as gospel, it is indeed at least semi-plausible that the saucer hull could be taken from a damaged or otherwise decommissioned starship of a given class and refurbished and installed on a starship of another class. (So, in theory, a Saladin-class destroyer could become a Ptolemy-class tug, or a Hermes-class scout could become a Constitution-class cruiser.) But if FJ's Tech Manual were to be considered canon, one would have to consider the ramifications to the structure of the STAR TREK Universe, namely that the content of the Federation charter and the use of stardates in that book is balanced heavily on references in "Whom Gods Destroy", indicating that the United Federation of Planets was only officially established when James T. Kirk was a young man, possibly after the Battle of Axanar. So all this talk of the Constellation's colorful past would likely be mooted.
If one takes any reference to the Tech Manual with a grain of salt, then the Constellation's origins and history remain a mystery. It is not clear what the ill-fated ship's registry means, if anything all.
This is just me, but over the years my thinking on the evolution of the Federation and starships has been evolving itself. Could the Federation and Starfleet seen in TOS have been the result of a slow and tumultuous diplomatic process that took place over decades, possibly beginning with ENT and not really fully forming until what happened with Kirk and Garth at Axanar? Maybe. If so, that might bridge canon of TOS, ENT and FJ.
Starships could've evolved over decades as well. I'd like to think that the Constitution-class of starships was the final stage in the evolution of the Warp 7 propulsion that Archer and Trip were toasting to in "These Are the Voyages..." Vulcan battlecruisers back then could only sustain Warp 7 for a short period of time. Perhaps a goal of the new Federation was to build and perfect a new class of cruiser that could sustain Warp 7 for at least that long, and each succeeding update of the cruiser design improved the performance of the evolving Warp 7 engine, which became known as true "warp drive" (a term never used in ENT, but suggested as a prototype in TAS "Time Trap"). When Pike ordered the Enterprise to divert to Talos IV at Warp 7, he made a point of announcing it to the whole ship as if it were significant; apparently the ship sustained that speed for some time to reach Talos. If the early Constitution starships were the ultimate evolution of the Warp 7 engine design, this would be a matter of pride.
As for the Constellation, if she was a member of an earlier fleet of vessels during the warp 7 era, and if the registries of Constellation, Republic, Valiant, Eagle and other ships were numerically tied to this (assumed) history, then those numbers may indeed suggest that Matt Decker's "lost" command was indeed a storied one, the result of at least one TMP-style refit (and possibly many) over perhaps several decades.