On another website I read a posters opinion of why we relatively rarely see federation member civilians, and so many Starfleet, even though Starfleet is likely only a fraction of one percent of the total population.
According to the poster it's because the civilians are irrelavent.
It probably the same with the members governments at different levels, for the purposes of the story all that's required is to refer to "The Council." There are only a hand full of mentions to the federation even having a President. We do know from the example of Ardana that member worlds do have functioning governments. In Journey to Babel, Sarek spoke of "my government," clearly not meaning the federation council, but the government of Vulcan.