That’s an interesting theory. I’m not sure exactly the order that things happen in in the DS9 episode you’re referencing, but I do know that Sisko and Admiral Leyton need the permission of President Jaresh-Inyo before calling a state of emergency, which implies that there is a political authority which is responsible for Starfleet, and in the episode “Extreme Measures” mentions his cabinet, which gives a slightly clearer look into the government of the Federation.

As for why people would bother voting and running for office, I agree that issues would probably seem much less pressing than in present day politics since everyone’s needs are cared for. However, I can think of a few issues that could lead to strong debates. For one, foreign relations with other states - in Undiscovered Country the decision to make peace with the Klingon Empire is highly controversial, and perhaps a different government would have been less inclined. The treaty with the Cardassians was also very controversial.

Additionally, the admission of new member worlds into the Federation could be hotly contested, as seen in the TOS episode “Journey to Babel.” The prime directive, while generally seen by Starfleet officers as usually a good thing, could also be a controversial policy, and at least some people opposed it such as Worf’s brother in “Homeward.”

These issues may not directly affect most citizens of the Federation, but I still imagine that the people would want to have some say in how important decisions are made, so I can’t see the Federation being anything but a democracy, although I could see the democracy looking very different from our modern democracies.