Deep Space 9 has an arc where a Starfleet officer attempts to launch a coup against the (elected) President of the Federation - we do see an elected leader there, but you're right that it isn't spelled out in much detail.

But is that elected leader elected by people or by representatives? This is where trek and wars meet. They both have a federal approach parralleling the US sennate, reps from individual planets rather than according to population. And the leader is elected by the sennate. So this isnt a government for people and thier rights, but a government of governments more akin to a United Nations with a navy. So there could be a nazi planet and a libertarian planet as how planets govern themselves is irrelevant. Individuals do not have rights under the federal government but through whatever planet they happen to be upon.

There are links to japanese fuedal system, which has a place in the US system whereby the federal government is allocated its own lands apart from the states. "The empire" would be as if washington DC and the US president started annexing state lands.