Well, the Enterprise had a mostly Earth-human crew, so maybe Starfleet was divided into different fleets under different planetary governments' administrations. The Starfleet we saw in TOS seemed very Earth-centric; maybe there were other subfleets that were Vulcan, Andorian, Tellarite, etc. The Intrepid could've been under the supervision of the Vulcan branch of Starfleet, for instance.
That theory goes a long way towards explaining why the Federation Starfleet would seem so species-specific during the TOS era, and also gives us some retroactive justifications for Kirk's referring to the Enterprise
as a United Earth ship and as being under UESPA command
From which one might draw the conclusion that, at least by the time of ST:TMP, Federation or Starfleet officials had begun to encourage more integration, a trend which continued into the 24th century; though it might be fair to say that the trend of having species-dominated vessels was a case of "old habits die hard" with many ships on-screen still having a predominantly human crew, or the all-Vulcan ship seen in "Take Me Out To The Holosuite" (DS9).
Taking another approach to that, I can imagine myself being an Andorian in the mid-to-late 22nd century, finding myself part of this new Coalition/Federation and thinking "OK: fighting along side Vulcans I can do, but serving
alongside them... no way!" I can see this trend disappearing over time, but it sounds reasonable to assume that even a century after the formation of the Federation, that it would still be "more comfortable" for individuals to serve with their own kind, but as time progresses, the barriers continue to break down and multi-species crews become increasingly common, and even encouraged (<cough>Titan