One question: Would we even know any of the "Constitution class" back story if it weren't for Star Trek/Lincoln Enterprises? Probably not.
Because, while "Constitution class" does appear in the Space Seed script and on a graphic created for that script (later used elsewhere,)* it was also appears to have been a production decision not to use that scene nor mention Constitution class anywhere else in the subsequent 55 episodes. Or include that bit of info in the writer's guide or The Making of Star Trek
To me, Starship class is consistent with the planned vagueness of the show: no actual dates, no specific century, no real speed measures, patches/colors for the uniforms, etc.
Just me 2 coppers; YMMV.
*(The Trouble with Tribbles in my mind places that graphic in a peer-reviewed journal: "...another technical journal?" "...time to catch up on my technical journals!")