Another school of thought is that in Kirk's time, Starfleet was chiefly an Earth organization (as was often stated onscreen), and the Intrepid crew could have transferred from Vulcan's space agency rather than came up through Starfleet Academy.
In
@Christopher's excellent 'Rise of the Federation' novels, he puts forth the suggestion that the early Federation Starfleet was made up of contingents from each of its founding member worlds, all of which are responsible for a different area of expertise. And that the different uniform insignia we always used to see in TOS didn't mean a different insignia for each SHIP, but rather for each of the "sub-fleets" that originally comprised Starfleet.
Meaning, these ships were all Starfleet, and crewed BY Starfleet officers, yet retained some semblance of their original mission dating back to the Federation's early years. To wit:
- Research & Development: Vulcan Space Council. Starfleet uniform insignia: IDIC symbol. Example: USS
Intrepid ("The Immunity Syndrome").
- Exploration: United Earth Space Probe Agency. Insignia: Arrowhead. Example: USS
Enterprise, USS
Discovery
- Military defense: Andorian Imperial Guard. Insignia: Pretzel. Example: USS
Constellation ("The Doomsday Machine").
- Operational support and supply: Tellar Space Administration. Insignia: Hoof. Example: USS
Antares ("Charlie X").
- R&D, similar to the Vulcans: Alpha Centauri Space Research Counci. Insignia: Rectangle. Example: USS
Exeter ("The Omega Glory").