As someone who earned a doctorate and earns some living studying territorial governments, I would not be surprised if an United Earth government had atrophied. Obviously, I like the idea that Earth would have a strong government, making its own laws, funding its own developmental projects, etc. However, Earth is effectively the capital of the Federation, and as such, the people would have a special relationship with the governance of the Federation. Government and administration would be local industries, just as they are in DC, London and Paris. People in these places don't struggle to be heard by the state: they have their hands on the levers of the state in many instances. It would seem natural that people in that circumstance would find some state functions to be redundant, even though they might be critical to other members. Would Earth have a militia? Maybe, but why make it a priority, either as a planet or as individuals, when Starfleet has a major presence on Earth? Why use an Earth government to organize projects when one can reach Paris by transporter? Yes, there's probably some form of government, but it would likely be weak when compared to the governments of Vulcan, Betazed, etc. The Bajoran Militia probably could kick the ass of whatever Earth force exists.
In the end, I would have found any attempt to show Earth government in action to be somewhat forced. I'm sure it's there, but it would contribute little,IMO, and its presence within the story should be perfunctory.