Apparently, some members are more equal than others.How equal the Member planets are in comparison to each other is unclear.
Couple of things.I'm rather pleased, by the way, that by Trek's time, they have gotten over this (IMHO, ridiculous) notion that the apparatus of national government has to be separate from any of its states.
First, it's not a given that the Federation building/buildings would not be considered separate from United Earth. The area around the Federation's collection of buildings could formally be consider "extra-territorial." Not only separate from United Earth, but legally "not on Earth."
Second, I don't see the Federation as a "national government," and United Earth and the other Member worlds certainly aren't "states." They're sovereign entities that have agreed on mutual co-operation in certain areas.
Remember the dinner scene between Kirk & crew and the Klingons in ST6. The chancellor's daughter mocked the Federation as a "homo sapiens' only club". From that scene and from everything else that I've seen, I get the impression that Earth is the dominant planet and that humans are the leading species and driving force of the Federation.
I don't know the reason for this. It could be that Earth has the strongest economy, largest population, best technology and/or that humans are the most assertive member of the Fed. And/or it could be something else.
Maybe that something else is that humans run Starfleet.
Also in ST6, during the classified meeting of Starfleet bigwigs, except for Spock, everyone else there was human, as far as I could tell. Earth/humans essentially own and operate Starfleet. It seems like Earth provides the infrastructure of Starfleet, the ships and almost all the personnel, the command officers and the academy.
Because Starfleet is the most important institution in the Fed, Earth/humans have the greatest clout within the Fed. Without Starfleet, the Federation wouldn't amount to much.
This is somewhat similar to the U.S. leadership position and role in NATO.
Is there a distinction between United Earth's Starfleet and the Fed's Starfleet? Or is it really a distinction without a difference?
On the matter of the Fed buildings on Earth, it could be similar to the arrangement between the U.S. and the U.N. regarding the U.N. headquarters in NYC.