San Francisco isn't the seat of government in the federation.
The government is in Paris. (Hmm... I'd say the European Union took over, but if that were the case, it'd be Brussels, not Paris.)
There seems to be some disagreement on that point. This site says that the President keeps his residence in Paris but conducts business from San Francisco. Wikipedia cites Star Trek IV as establishing the legislative branch of the Federation in San Francisco, but the executive branch in Paris. (I haven't watched the movie lately, so I don't know how accurate that is.)
Hmm. Well it's hard to disentangle the exact details without knowing the specific sources the material comes from.
I have reasonable recall and easy access to all of TNG and DS9, but anything established outside those two series is harder to verify from original sources.
Of course, not all such information comes from the show itself, which leads to long and tedious debates about what constitutes valid 'evidence' for a fictional universe. (I used to deal with debates that dealt with cross-overs, which forces you to be very specific in what does, or does not officially belong in any given universe.)
It's possible I got it wrong, because this isn't something explicitly mentioned all that often.
You see depictions of Paris and San Francisco on a regular basis, but it's much easier to find a reference to where starfleet academy is based than to the government itself.
A quick search seems to imply a split between novels and the shows themselves.
Also, the most direct evidence would seem to be from the original series and Voyager, which I'm the least familiar with.
Lol. That's the trouble with knowing too much about something, but not everything there's always room to be wrong.
I suppose it just made more implicit sense for a government that spans the whole planet (and more), NOT to be centralised in one specific country.
(Take the closest real-world analogy to the federation - the European Union... The Parliament is in Belgium, and the high courts are in the Netherlands. It has no military, so that doesn't say much. Of course, an immediate counter-point is how close these two things actually are to eachother. And the fact that they're located in 2 of the 3 countries that originated the idea behind the European union. Before it existed, there was the Benelux. Which had open borders, and several of the other characteristics of the European union. This consisted of The Netherlands, Belgium, and Luxembourg. So... The location of the centre of the European Union is based on it's origins, more than any other logic...)
Anyway... Since the only thing I can easily find that explicitly says the government (and not just the president), is in Paris, are novels (which aren't generally considered to count), I guess I have to retract that statement. XD.