Our interpretations just vary.And again, "UESPA became the Federation" is a nonsensical statement. How can a Space Probe Agency become an entire nation? That's like saying the DMV became the State of Ohio, or that the British Museum became the United Kingdom. It's utterly meaningless.
United Earth government is a member planet in the Federation (like the USA is a member of the UN). UESPA was the United Earth government organization put in charge of its space program. The space program's command and control was called Starfleet under UESPA. At this point, Earth starships reported to Starfleet Command, which reported to UESPA, which reported to United Earth government, which was a member (but not under any real obligation) of the Federation. The Federation council would request starship support from various member planets. In United Earth's case, if they decided to help, then they ordered UESPA to send a ship. UESPA would order Starfleet Command to send a ship. Starfleet Command would order a starship to go someplace. A real bureaucracy. In the meantime, United Earth is sending its starships out without any real coordination with the Federation or any of the other member planets.
When United Earth decided to slimline the bureaucracy, improve coordination of space assets between other planets and beef up the military arm of the Federation (probably due to the new Romulan threat), United Earth dissolved UESPA and transferred Starfleet Command and its assets to the Federation and also put the combined starship assets of all member planets under Starfleet Command. Starfleet Command now reports direct to the Federation and member states are to provide military assets based on treaty agreements. During the transition of Starfleet to the Federation, starships in the field were still trying figure out "who's on first."