Why, exactly, do you consider TMoST to be the end all source for TOS? I'm asking more from my own curiosity than anything else, and I would agree it's a very interesting reference and insight into the development of the show (I have a copy myself). But I'm also aware that it was published while the show was still in production, meaning some things referenced either didn't come to pass or changed over the course of time. It may be true that no specific reference to the Enterprise being a Constitution class ship had been created at that point, but that doesn't inherently prove later associations to be wrong.
I think the whole "starship class" thing is perhaps being taken too literally, and within a narrow context. In the context of the production, when it was not very clear what other ship categories would exist besides that of the Enterprise (or indeed if it were possible to even see other ships occasionally, owing to budget issues), then it makes total sense to use "starship class" as a vague descriptor of what makes Enterprise unique for storytelling purposes. It would be like if I were in a naval movie, and I referred to an Arleigh Burke as a destroyer class for a general reference. If the script required it, I could actually say "Arleigh Burke class" or mention that it's a guided missile destroyer, but that doesn't inherently matter unless it fits into the story somehow.