So we have a multitude of opinions and two parties to represent them. Does that mean that they are or are not "representative" of the people? I'd say yes and no. The two parties certainly do not do a comprehensive job of representing everyone's opinions, but they do a fairly decent job of representing those in the middle of the ideological spectrum. Though we often like to think that, for any given issue, there's a "republican" position and a "democratic" one, but the truth is that there is diversity within each party as well as between the two parties.
That's because people are different. Different people have differing life experiences that have led them to have a different world view than others do. Simply having a polite discussion isn't going to change the fact that these differences can be huge. Then add in fact that people are not quite the rational beings we like to imagine we are and that leaves me with little hope that there is a simple way to resolve difficult and troubling differences amongst the various groups of people.