The TNG crew should have been more diverse in the first place, you are sending out your flagship of the UFP, the USS Enterprise and its mainly 99% humans? Really?
Of course we all know that this is a mostly Doylist decision for TOS, to save money, and part of the reason TNG/DS9 are more diverse is because the effects were cheaper and more re-usable (and the budgets higher). But on a Watsonian level, it suggests an evolution in the nature of the Federation that I find really interesting.
There's a lot in TOS that suggests the Federation is fairly young and the individual member systems are still very independent - not just in political power but in personal and collective identity. Even in Star Fleet people identify very strongly with their homeworlds, and "Journey to Babel" strongly suggests that most people identify with their homeworlds first and the Federation second. The Federation isn't a single polity; it's a bunch of mostly indpendent polities more-or-less working together.
By TNG this has changed; the Federation is the dominant polity and identity and whilst people still identify with their homeworlds, the "I'm from the Federation" component is much, much stronger. In TOS, it's more like the EU where most people think "I'm French" or "I'm German" or "I'm Polish" first, and "I'm European" second. In TNG, it's more like Canada and the US where most people think "I'm Albertan" and "I'm Canadian" side-by-side.
Anyway, this makes me feel that the various starships are
all 99+% one species.
Enterprise is almost all human.
Intrepid was all Vulcan. Presumably there are all Tellarite and all Andaroian crews too. This could be formal, as an exercise in politics so every member feels fully represented. Or it could be practical - it being a lot easier on life-support if everyone aboard likes the same light levels and temperatures. Or it could be unthinking, with human captains mostly knowing other human officers and so collecting crews that are mostly human. But it does seem to happen.
(I'm sidestepping the whole question of "are they all actually human?" but it's also worth remembering that TOS encounters a whole slew of non-humans that are outwardly indistinguishable from humans)
To loop back to the OP, I think how a Vulcan captain in charge of humans would behave depends a lot on the era. In TNG, there's clearly a lot of edges rubbed smooth. Ordinary people still have friction, but there's an awareness that differences exist and they aren't inherently bad, just different. In Star Fleet, as others have said, presumably there's training at all levels on how to get along with the many other cultures you'll be working with.
In TOS, I think there would be serious friction, just as there'd be serious friction with a human commanding vulcans, ultimately ending with either the captain being reassigned or the captain adopting behavours consistent with the crew. But I also think that in TOS this would be a rare and probably temporary situation, because there's a consistent suggestion that all the ships are single-species, and
Enterprise is weird for not being so.
Obviously, this is all just my opinion and interpretation of the material, and not some irrefutable fact.
