Howzabout this: Part of the responsibility of member planets, at least those of a certain size, is to maintain a certain number of Starfleet vessels. This would not only account for Kirk's UESPA citations, but would also for the Enterprise's almost all-human crew, just as it would account for the all-Vulcan crew of the USS Intrepid; their operating authority was probably some branch of the Vulcan Science Academy. Somewhere, there's probably a starship with an all Andorian crew.
If you think about it, it makes a lot of sense to have your crew predominantly of one species. You only have to set your shipwide environmental systems to accommodate one standard, with any individuals from other species with different environmental standards only having to worry about their own quarters. So, the common areas of the Enterprise can be set to 72 degrees, while Spock can set his own quarters to a nice balmy standard Vulcan spring temperature of 120. By the same token, the shipwide temp of whatever ship the Andorians have can be set to a nice comfy 45 (comfy for Andorians, at least).
This is much how I reconciled the UESPA / Starfleet / "only twelve like it in the fleet" issue. Think of the Federation Council as "the Club" non-aligned alien worlds and colonies of Federation member-worlds are not directly involved in. How does one join "the Club"? I wondered about that for some time. Given the procession of obnoxious bureaucrats and political personalities we've seen, it's obviously an insular, "you can't play in my sandbox if I don't let you" clique. "The Galileo Seven", "A Taste of Armageddon", "Metamorphosis", and "The Trouble With Tribbles" should make that clear.
So, how does one join "the Club"? Well, we've seen from TNG's 4th-year ep "First Contact" that warp activity becomes the threshold for basic interaction between the Federation and alien societies. Pre-warp societies like the "Roman" inhabitants of 892-IV and Zeon/Ekos are handled more gingerly by the Prime Directive. But when you're ready to join "the Club", you're well beyond Malkor III. You don't just have warp drive and open trade relations with the UFP, you're ready to proove yourself as a viable member
and a
major player/contributor. I doesn't take an Einstein to see that the Enterprise was being refit/rebuilt for TMP in Earth orbit for an obvious reason. She's a UESPA-sponsored Federation Starship. That makes her Earth's pride-and-joy well
before she's a Federation flagship.
So, here's how it would have to work: each Federation council member would have to prove itself by sponsoring a number of Federation starships, all built to Starfleet specs and prooven in trail runs. Enterprise, and quite possibly Lexington, Defiant, and maybe Constellation, Excalibur, Hood and Potempkin, (and presumably others) would be UESPA-sponsored, meaning it's the responsibility of United Earth to build, test, and predominantly man each of these ships to validate Earth's role as a player. Spock's presence as the token Vulcan was likely a political decision. He's not a spy, but he is there in addition to his duties to report on the ship's normal operations per Starfleet regulations. The would explain why he's science officer, XO and a walking rule book.
It would also make sense that Federation starships would have crews of similar species, if not each one being only a single species, since that would make it easier to set up the Sickbay and medical staff to care for everyone the same.