Council democracy is not a bad idea per se, after all they shortly had it in France. I think that worker participation in the work place is very important if you wanna have true democracy but on the political level I am a big friend of centralization. It is just not possible to use this spontaneously creative, direct democratic council idea on population levels of millions in the case of nation states, billions in the case of world states and trillions in the case of the Federation council.
So while the Federation has probably little to do with American style capitalist democracy which is not a system that is stable in the long-run due to the conflict between capitalism and democracy it also has most likely little to do with council democracy due to the large population of the Federation.
Democracy may have a far different meaning in the 23rd century. It really is a meaningless word to the viewer without knowing nuts and bolts of how it works in an interstellar alliance.
If I'm not misremembering, a world has to be unified to join the Federation. Nothing about how it is unified.