When the Enterprise went back in time, it would suddenly appear in its "experienced" state and disappear from where it was the first time around. That ship would vanish.
No, but meeting up with themselves would have been a paradox since it did not happen "the first time around." And that's the real crux of the matter, and where many people (including authors) get tangled up in time travel stories. The events of the universe do not "happen again" just because a time traveler folds back over his own path. As Doc Brown said, "You're not thinking fourth dimensionally."
(The only way to have the Enterprise
meet up with itself and lead to a different chain of events without it being a paradox is to invoke a multi-verse. This is the flaw in BACK TO THE FUTURE—Marty interfered with known events multiple times, which means he could not have known about them and the need to "correct" them. That's a paradox. So BTTF must be a multi-verse... but then nothing any Marty in any universe did could lead to the one on stage playing guitar to vanish. Double flaw. Meanwhile, BTTF2 had a pair'o'docs handing himself a wrench.)
"Tomorrow Is Yesterday" does not work no matter how you slice it, while "Assignment Earth" is a perfect example of a reflexive causality—meaning the Enterprise
did not "change" anything, as they were a part of that history. They simply didn't know it going in.