Maybe there is something that contradicted this that I'm not thinking of (in TOS), but aren't there a few references in TOS that Spock is the only non-human on board? I'm thinking of McCoy complaining about having to have special medical setups for one person. (As opposed to McCoy just complaining.)
This was due partly at least to time and budget, I realize. But it also lent to the feel in TOS that humanity (and the other planets as well) was still kinda new at this. You know, "frontier"?
This has gone by the wayside in later Trek and because that's what we're used to the bulk of Trek looking like it has seeped back into present day 23rd century shows. Even TAS promptly added Arex and M'Ress.