Actually, the name "Constitution class" came from the 1973 Technical Manual, because the official dedication plaque couldn't be read on-screen due to size and grainy film of the 60's, and the tech manual listed a NCC-1700 USS Constitution, which, being the "first" of the line, would have passed it's name on to the entire class in naval tradition.
However, with remasters, photographic enhancement, and the actual plaque from the set, the official title "Starship class" overrides the previous "Constitution class". Many trekkers have called that design "Constitution class" for so long (I have all my life) that suddenly switching to "Starship class" is a big leap. To prove my point, the designers of the Defiant set in Enterprise were so used to "Constitution class" they put it on Defiant's dedication plaque.
Furthering the confusion, there may be "non canon" sources that pegged the post-refit (Movie) Enterprise as the "Enterprise class", again, a name carried throughout various parts of lore, even though many still called the movie ship a "constitution class" due to the carryover of design aesthetics. The NCC-1701-As plaque does nothing to clarify the situation, it has no class identifier on it at all.
So, depending on who you're talking to, what they're used to, and which version (TOS or TMP) being addressed, you can get one of 3 "class names" for the same ship.
For sake of "clarity", I'll stick with the "constitution" designation for TOS-era, and "Enterprise" designation for TMP-era, to help differentiate the technology packages (and my own convenience, having used these terms for 20+ years).
Now, the framework/keel that makes up the original Enterprise is 20+ years old, and, considering the refit, frequent stresses Kirk put on it, and all the holes Kahn put in it, that framework probably was more suitable as a museum piece or scrap metal than repairing it (more like rebuilding it again). Hence the decision in ST III to decommission the Enterprise.
However, at the same moment, the "Enterprise class" design was only 5 years old (best guess from TMP - ST III without research), and the Excelsior class was just entering the testing phase, needing at least a year of testing and another year of construction before the next Excelsior class ship hits space. So, the "Enterprise class" would have been in active construction, not "going out the door", which is why Kirk gets one of them, new, as NCC-1701-A instead of waiting 2+ years for the next Excelsior class.
With that fact (Enterprise class variant of connie/starship class being in "late" production), and the fact that all 3 classes (Enterprise, Miranda & Excelsior) came out around the same time, it would not be a stretch to see them in TNG time, especially since it's proven that Miranda and Excelsior designs made it all the way to TNG.