After WWIII a lot of the disgruntled displaced old governments would have made some powerplays.
After the fall of the United States of America, don't you think that the Native Americans would have thought about getting together and kicking whitey out?
From DS9 Rapture.
WHATLEY: Not sorry enough to return my comm. signals. All three of them. That could get an officer in a lot of trouble. Look, Ben, I need to know that I can count on you. Now, Bajor's admission is only the beginning. Now comes the hard part. Federation council members have to be chosen, the Bajoran militia has to be absorbed into Starfleet. There are thousands of details that have to be overseen and you're our point man here. That means we need to depend on you more than ever.
Chosen, not elected. Slicing the Melon finely I know, but the text suggests a political appointment and not an elected position.
I google searched every Star trek script for the word Democracy, and other than a mention of "the dark Time" in Klingon history when they experimented with it, every other mention of Democracy is the common idiom "This is not a Democracy" some Captain belts out when the little people try to be difficult. Using the same proccess the onlymention of elections are about Kai Winn and Shakaar from DS9 so, really?
The Federation was Founded on Pinko Ideals.