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 during TOS episodes produced before they came up with the idea of the Federation and the Federation Starfleet.
Who knows? Maybe when Kirk said there were "only twelve [starships] like it in the fleet," he was referring to the Earth fleet, which was anchored by the Constitution
-class starships. Maybe there were other fleets with different classes of ship. Since we never saw the Intrepid
, maybe it was a Vulcan ringship, one of the capital ships of the Vulcan branch of Starfleet.