In the 23rd century Federation and later in the 24th century Federation as well, there appears that there are a relatively small number of heavy, influential power players, Earth and Vulcan being two predominate examples. If the Federation were simply a "Earth Federation" (or Empire) then the events of Journey to Babel wouldn't have occurred, Earth would have just rammed through their choice on the matter, no conference, no assemblage.In TOS the implication was that the Federation was really the "Earth Federation", that humans invented Warp Drive all on their own and merely invited other alien species to join them as weaker partners in the Human organization. In other words it was a benign Terran Empire, not a true multi-species cooperative.
I never received the impression that the Federation was a Earth Empire, but I did get the distinct impression that Earth under took activities separate from the Federation and that the Enterprise was one of the ships that engaged in those activities. Also that Earth had it's own bases, outposts, colonies and interests, again separate from the Federation.
You can retcon it all you want, but that doesn't change the fact that Kirk frequently referred to the Enterprise as a Earth ship. That's in-universe.
It kind of where I arrived at the Federation really not being a true government and instead a interstellar organization, or to borrow Anwar's phrase, a "multi-species cooperative."




