Starfleet was still solely under United Earth which is a human based organization up to the mid-first season of TOS, when it was transferred to the Federation (
when did TOS take place, 23rd century or 22nd century). Aliens (and half-aliens) are rare, or should be rare, serving with humans in the CAGE, DSC and still rare in TOS. Long-term
environmental (and language) concerns probably limit most alien postings to earth based starfleet and ships. Even Spock, who is half-human, finds the Enterprise uncomfortably cold as he finally confesses to McCoy in "The Deadly Years". Vulcans will find it too high in O2 or high pressure, and too cold (per "Amok Time"). Andorians will find it too hot (ENT retcon). Tellarites too...clean smelling or too high gravity judging by how weak and unstable the ambassador looked in "Journey to Babel". Short-term, you can tolerate it, but long-term can be, if not uncomfortable, hazardous to your health.