Nothing in Star Trek ever suggested that the United Earth government was forced upon the people of Earth. On the contrary, there is every suggestion that its existence was the will of the people.
You have eloquently stated Gene Roddenberry's biggest fantasy: that in the future, all of mankind would adopt the religious, economic, and political preferences of Gene Roddenberry.
Then there is no need for a squabble, because every last person on Earth has wholeheartedly joined the same single faction.
not necessarily, I don't think it's "utopian" that in a society with access to abundant resources, that has eliminated poverty, and where high-quality education is available to all, that there wouldn't be a large faction clamoring for a return to unrestricted capitalism and massive exploitation. There may be dissident groups like the Maquis from time to time that reject the political order, but I doubt it would be very common.