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.
We'll have to disagree on the meaning of "direct evidence". Since it got cut
, to me it no longer is "direct evidence" regardless of the reasoning behind it. At best, it falls under a "what if they had tried this?" or "they thought about doing this but at the last minute changed their mind" backstage scenario.
We can of course argue that in later years they revisited this idea and made it "evidence" by inserting it into "The Naked Now" episode and later movies (with the exception of "Wrath of Khan") and series (and I'd agree).