We have evolved since then.
Have we? Was not that long ago that we fire-bombed Dresden. There is still a major controversy going on today over Gauntanamo. Not saying I don't support our Nation or what our government does, I'm just saying that people need to get past the idea that America is incapable of doing wrong.
but enough of politics, lets keep this within the realm of Fallout.
Much like the Enclave, I am saying they arent America because they werent ACTUALLY part of the government as far as I know. They were an organization, whether they had 'ties' to the government or not, it is purely self proclaimed that they are the American government. Forming your chain of command around the government, using propaganda and a similar flag does not make you America. It makes you a copycat
They weren't actually apart of the government but the President of the United States went to the Oil Rig before the great war? Even when the senate apparently did as well?
Even if we assume that the Enclave was a seperate organization from the government of the United States, it seems pretty clear that they were the ones in power, and thus this argument falls into play:
But now on to your subject about where a government derives its legitimacy from. As an American, I know that we have the idea that a "governments power and legitimacy is rooted in the consent of the people." (Social Contract Theory, John Locke) The problem with that idea is that it is a "relatively" (and I stress relatively) new concept (yes a form of this was used by the Greeks and Romans but they both soon fell into monarchies anyway so...). For most of history, governments did not need the "consent of the people" they only needed the power and authority to enforce their will. What you defined is the "American/British" definition of the word "government" (British/American when it was first developed anyway), but in reality, no matter what definition you go off of, the people in power are "the government" de facto. Dictators do not need the consent of people voting but they are "the governent".
Even if the officials in "the Enclave" were not voted in. They are still the government, because in reality, governments do not need the consent of the people, there isn't some "outside" power that says "HEY! YOU NOT LETTING PEOPLE VOTE, YOU NOT GOVERNMENT!" If an organization takes power and hold the real authority and makes the decisions in the country, it is the government. Period, no matter what anyone says.
If America were to descend into a monachy today, would the King of America be our government and head of the country? Yep. Even if he didn't let us vote? Most certainly. All hail his majesty.
The Enclave didn't just say "HEY WE'S AMERICA NOW GUYZZ!" and then make up a "cool" new flag and such. They are the continuation of the
Fallout American Government, which was already very different from our modern day government
pre-war. The problem is that people are looking at the Enclave through the lenses of "present day" glasses.