Re: Amok Time doesn't make sense.
True, and remember most times the Enterprise was mentioned as being a "Earth ship." Spock was apparently the only alien on board and there was no indication of "founding members" of the Federation. And Starfleet seemed to be Earth-centric (United Earth Space Probe Agency, for example). Most of what fans know about the Feds and the Vulcans came much later in the sequel series.
Of course in the "Earth Ship" and UESPA episodes there was no Federation. The Enterprise was an Earth ship. Star Trek was still evolving back then.