Spock appeared to be the only nonhuman assigned to the Enterprise, so she'd be super conspicuous.
Maybe not the
only one, but with few enough others that they'd probably all be known. It seemed in "Charlie X" (TOS) that there was at most one other besides Spock...
CHARLIE: How many humans like me on this ship?
RAMART: Like a whole city in space, Charlie. Over four hundred in the crew of a starship, aren't there, Captain?
KIRK: Four hundred and twenty-eight, to be exact. Is there anything we can do for you, Captain? Medical supplies, provisions?
CHARLIE: Hundreds. All human, like me. That's exciting. Is that the right word?
(Kirk didn't customarily seem to count himself among the usually-quoted figure of 430 crew, IIRC.)
But of course, that was merely
at that moment. There could have been others who transferred aboard subsequently, as likewise there could have been others who'd served aboard previously and had transferred off. Maybe no Trills, though. (And maybe there weren't even any in Starfleet at this time; we don't really know. I don't
think we even know whether they were UFP members yet...or ever, for that matter!)
Sisko got away with explaining Kirk's not knowing his face by saying he had just transferred over from the
Lexington, a situation which cropped up a time or two elsewhere in TOS as well, but it would undoubtedly be best to take as few chances as possible of being noticed as out of place until the mission was accomplished.
IIRC, the first Trill to serve in Starfleet - the first Dax host to do so, anyway - was Torias, and he wouldn't come along until some time after TOS. IIRC, Torias was Dax's host was during the 2280's.
I don't believe that's from any episode. In fact, I don't think Torias was ever said to have served in Starfleet at all, let alone to be the first Trill (or Dax) to do so. Could be from a novel or somesuch.
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MMoM