Actually, the franchise never makes it explicit that Star Fleet is an arm of the Federation (UFP). However, in the Enterprise series (STE), it is explicit that Star Fleet is an arm of United Earth Governments (UEG) which goes on to become a founding member of UFP.
The existence of a version of a Starfleet (that is in some way related to the UESPA) prior to the existence of the Federation complicates the issue, but it doesn't contradict the notion that Starfleet is principally under Federation control (though at least through the 2200s, taskings by the UE were possible, even outside of UE territory).
For the purposes of Star Trek Hunter / Star Beagle Adventures, I felt that UEG was no more likely to turn Star Fleet over to UFP than the US is to turn the US Navy over to NATO
That's not really a valid comparison.
NATO is a non-governmental military coalition, not a government. State assets of various sorts (inc military) are made available to the federal government on a regular basis.
This assumption is supported by the original series episode The Immunity Syndrome, in which Kirk mentions that Star Fleet was experimenting with an all-vulcan crew for one ship. All-human crews are fairly close to normal.
But is contradicted as early as The Gallileo Seven with Federation High Commissioner Ferris having authority over Kirk for the mission that they were on, and the TOS movies are pretty clear on the point that Starfleet is under Federation control by that point.
Also, the idea that the all-Vulcan crew of the Intrepid was an "experiment" is fan speculation (conversely individual Vulcans on mixed-species Starfleet crew has been explicitly described as such) and is not supported by the text of the episode. The most that we can conclusively say is the Vulcan unlike Earth doesn't appear to have gone "all in" on Starfleet and maintain their own starships.