I stand corrected: "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 film editor's omission of evidence in "Space Seed".)"
The omission (for whatever reason) unfortunately took away any direct evidence of the Enterprise belonging to the Constitution class in TOS. Obviously, the later series confirms it but that is outside the OP's original post...
Well, it didn't take away any
direct evidence; the script directions
are pretty, well, direct
. And fortunately those
weren't taken away.