I'd agree that the Enterprise could have belonged to any ship class, even her own, due to the lack of direct evidence (especially after the director's omission of evidence in "Space Seed".)
Well, just to clarify: it doesn't seem to be an omission
by director Marc Daniels. It would seem that "Constitution-Class" was scripted, graphics were made, and then the scene was indeed shot by Jerry Finnerman under Marc Daniel's direction. Any later decision to not use the "Constitution-class" graphic (if, indeed the used
"exploded wing" graphic is someday found to not
say "Constitution-class" on it) would appear to be an editorial choice made by Film Editor James D. Ballas. It was probably just due to a real glarey screen--as opposed to some kind of outrage that the Constitution
was being identifed as the class ship.