I think Basil hit the nail on the head. We already live in a world where many of us have jobs that require us to serve many different masters, and where we must answer to many different authorities. It's not at all likely that things will be all that different in the 23rd Century.
I have no problem with a Star Trek universe where both Starfleet and UESPA coexist, along with a whole slew of other galactic and regional institutions and authorities that Kirk must sometimes answer to as his ship travels from one region of space to another.
Perhaps on those early occasions when Kirk asked Uhura to contact some authority other than "Starfleet" (Such as "Space Central" or "Space Fleet Command"), it was simply because that was the official body that managed affairs in the particular region of space that the Enterprise was currently traveling through. Later in the series, we can assume that the reason that these regional authorities seemingly dissappeared, is that Federation influence was expanding, and more and more of these regional authorities were being replaced by, or absorbed into, Starfleet and the Federation itself.
I have no problem with a Star Trek universe where both Starfleet and UESPA coexist, along with a whole slew of other galactic and regional institutions and authorities that Kirk must sometimes answer to as his ship travels from one region of space to another.
Perhaps on those early occasions when Kirk asked Uhura to contact some authority other than "Starfleet" (Such as "Space Central" or "Space Fleet Command"), it was simply because that was the official body that managed affairs in the particular region of space that the Enterprise was currently traveling through. Later in the series, we can assume that the reason that these regional authorities seemingly dissappeared, is that Federation influence was expanding, and more and more of these regional authorities were being replaced by, or absorbed into, Starfleet and the Federation itself.