That's not really the strongest evidence. To another viewer, it could be so obvious that she's a genetically engineered cat (or whatever)
Why would they believe that so obvious, when up to that time, ST never said or even hinted that Starfleet has any genetically engineered personnel serving? That's just fanboys being too obsessed with trying to provide an answer to a question that no one felt important enough to ask...for obvious reasons.
that, if she were really an alien, they'd have mentioned her planet name, her species name, and so forth.
No, they would not, as everything did not need to spelled out to the audience...like later version of ST. It is a far future sci-fi series where ST--at the time--featured creatures from all over the universe. Most did not have some explanation of origin, but viewers accepted it, as it fit within the genre-natural frame of a populated universe.
If, after all of that--someone is still confused or not convinced, then they need to watch
Enterprise (good luck) or
Dragnet, where someone will spend half the running time telling you too much information about things that are apparent.