An interesting theory if not for the fact that it's based on a faulty premise.
Well, TOS has all sorts of inconsistencies when it comes to terminology, especially first season. What stuck out in my mind more were the times that people would ooh and ahh over the fact that the Enterprise wasn't just any ol' ship, it was a STARSHIP
Regardless of whether the term was unique to the specific model of ship that the TOS Enterprise was, it's still a working theory that in this case, "Starship class" was being used as a generic term, e.g. "aircraft carrier", "battleship", etc., as opposed to the name of that specific class, e.g. "Nimitz-class aircraft carrier".