I'm thinking of what was the state of the world when the government was founded? Post WWIII. Then some tall skinny nut job comes out Montana and says "Hey, I discovered faster than light travel! Oh, and the ALIENS are here!" (I'd put that in the column for "ZC really discovered something that the neighbors didn't have" because warp drive or no, why are the Vulcans making first contact with a world that JUST bombed itself into oblivion? Screw you, ENT. But that's another thread.)
What governments there are are going to pick themselves up as best they can and then surrender their sovereignty to this new governing body? Yeah, they're not doing that over the phone. OR it's not the existing governments but a new movement doing an end run around the powers that got them into this mess. Also not over the phone and probably in as neutral a territory as can be found. (Hello, penguins!)
How does ZC's flight in 2063 square with Q's post atomic horror of 2079 "by which time more rapid progress had caused all United Earth nonsense to be abolished"?
I don't think it's particularly unrealistic to imagine that things might have gotten worse in the 2070s before they got better. "Attached" (TNG) establishes that the last of the independent national governments agreed to join United Earth in 2150, so that means there's 87 years between First Contact and the final unification of Earth.
The novels set in the First Splinter timeline had previously established that the Traité d'Unification establishing United Earth was first signed in 2130 (
Articles of the Federation).
Section 31: Control established that a prior attempt at a united world government called the United Earth Republic had been established in the 2110s but had collapsed before the Traité was signed. So the unification of Earth was a process that took at least 50 years, with starts and stops, and with the sovereign state called United Earth apparently existing alongside national governments that had not yet joined it for at least twenty years. Of course, that's all the First Splinter's history; actual canon might end up different if they decide to establish anything about this stuff in new episodes.
Geneva has been my "headcanon" (or whatever we're calling it these days) Earth capital for a while now, after I read something (unlicensed, IIRC) that claimed that. (I don't know for sure, but I would guess they picked Geneva due to Switzerland's famed neutrality, plus the well-known Geneva Conventions.)
Myriad Universes: A Less Perfect Union by TrekBBS's own William Leisner establishes Geneva as the capital of United Earth in at least one alternate timeline. As you say, this is because of its famous role in international relations -- site of the signing of the Geneva Conventions, home of the United Nations Offices at the Palais des Nations, and referring to Swiss neutrality. (Geneva is also the capital city of the Earth Alliance on
Babylon 5.)
I would agree with this. As someone who lives in Ontario, the thought of a single city being both the national capital and a sub-national capital seems such an odd concept. So I don't like the thought of Paris being the capital of both the Federation and Earth. I haven't looked into it, though, so it may be there are other places in the world where this is done, so it probably wouldn't seem odd to those living there.
Something somewhat similar happened to the United Kingdom, since London was famously the capital of the Kingdom of England before the Acts of Union.* It's not exactly the same, of course, since the U.K. is a unitary state and there is no devolved English government the way there's a devolved government for Scotland, Wales, and Northern Ireland.
In the Swiss Confederation, there is no
de jure capital, but the City of Bern is the
de facto federal capital and is also the capital of the Canton of Bern.
But yeah, the more common scenario appears to be that either a federal union makes the federal capital a special district, or makes the federal capital into a constituent polity in its own right (e.g., Berlin being both the federal capital and a state in its own right).
*
Okay, technically the City of Westminster located within the large urban settlement commonly called London, itself named after the City of London which neighbors the City of Westminster. But I digress.