For sake of argument, I could accept "Star Ship Class" as the official starship class nomenclature of the Enterprise and her sister-ships in TOS, but there are a few major hurdles to clear:
1: Kirk said "there are only twelve like it in the fleet" obviously implying starships like the Enterprise.
2: Judging from the list of starship names I compiled upthread, it seems very strange indeed that a Federation of multiple worlds and species would have only a very limited pool of starships, all with Earth-derived names.
3: This TOS-only-cannon approach put TOS in complete isolation. The original Enterprise and her sister-ships would truly be "in the freezer". No Reliant, no Mirandas or other ships loosely related to the Enterprise at all. Just a very small number of "Star Ship Class" vessels and that's it.
4: The only way I can see adhering to this strict orthodoxy of TOS-in-isolation continuity is if you also agree to the strong implication in "Whom Gods Destroy" that the Federation only came into being after the Axanar Peace Mission, in other words, when James T. Kirk was a young man. For a multi-world Federation to have only a dozen or two dozen "Star Ship Class" vessels in her armada, the Federation would still have to be pretty young anyway.
This is a very interesting exercise in fandom imagination, and could occasionally be useful, but purely academic at best. One could very logically argue that from TAS or TMP forward the entire STAR TREK franchise has become an exercise in concept erosion ultimately resulting in the 2009 movie. I'm not a fan of the 2009 remake, and I regard it as more of a light-hearted spin-off of the franchise (for lack of a better term). But I don't think it's necessary to isolate TOS entirely in order to insulate it from the straying effect of Hollyweird. (TOS had plenty of follies and built-in contradictions anyway.)
Any sci fi dramatic franchise is going to have an experimental nature to its content, it's going to stumble from time to time; especially one that pioneered the concept of an hour-long scifi dramatic series on primetime network TV. That's the nature of the beast.