Original Star Trek is hard to reconcile with later Trek if you refuse to accept later Trek as being fact (or at least, the most accurate interpretation of that information). For example, it was mentioned on this thread that Star Fleet doesn't refer to an organization, but to a star fleet. However, in later trek, Starfleet is referred to as an organization, and that the original series Enterprise was a part of that organization. Also that the Federation Starfleet has always existed.
I know we try to desperately find explanations to make original Trek fact, but later Trek contradicts it. Its simply that. We know that UESPA is not giving orders to Starfleet by the end of the 23rd Century. We also know that non Federation governments view the Federation as Earth. Earth is the capitol of the Federation, and what we have seen on screen is that humans make up the most of the Starfleet's population. We also know that everything is in English on a Starfleet vessel (although I'm sure panels can display in any language, that doesn't change the fact that signs, like on doors are in english). English isn't an alien language. Its a language from Earth. Where the Federation has all of its government buildings.
So for me, which of course is just my opinion, is I disregard the UESPA references in the first series. It makes no sense to the greater continuity of this universe. Starfleet is the Federations defense agency, and is responsible for all defensive and offensive operations. At the same time Starfleet is a large organization, and performs many peacetime tasks. Like exploration.
Later trek has inserted UESPA into Trek history. So it did exist.
Some agreement, some not.
IMO, "Starfleet" is Earth's organization and always has been, but Earth -- being a Federation member -- has to obey the edicts of the Federation just like everyone else. I think people are grossly overstating the cohesion of a multi-planet superstate like the Federation professes to be; it's not just "America, only larger." This is a state where individual members constitute whole solar systems
. Nothing even comparable to that exists today; with our current level of technology it would be taking every country, every state, every province and territory in the world and putting each one on its own little planet, all orbiting the same distance from the sun. In that scenario, the United States wouldn't share the same military or the same space program; it would be miracle if they shared the same CURRENCY.
I think that Earth just happens to have one of the most prolific and sophisticated space services in the entire Federation. It's a bit like how the Russian Federation and the United States have/had the only workable manned space systems in the world and therefore all the contributing members of the ISS depend on the Soyuz or (until recently) the space shuttle to get astronauts and materials to and from the station. Europe has an unmanned cargo ship that's very efficient at sending supplies, Japan has an effective if slightly less capable one. The SpaceX company now has one too, but at the moment the Russians have the best space program for manned flight and so anyone who wants to do anything in Space these days goes and works with the Russians. I see Starfleet as a similar organization, just diversified to the point that they can do just about ANYTHING in space, up to and including kick ass. Other Federation members almost certainly have their own fleets, their own ships, their own priorities, their own command structures, but like Starfleet they're subject to the overriding authority of the Federation, and UNLIKE Starfleet, their capabilities are so limited that nobody in their right mind would ever send them to do anything in deep space.