Agreement, and conclusion should be reached by applying procedures to which sides* agree: be it a coin tossing, asking oracle, or voting. Opposing side won't magically disappear when your side achieves power over media. What will likely happen instead they will get more reasonable motivation to wrestle this power from your hands with any means possible.
*Btw, usually there are more then two sides: it's just stale majoritarian system which leaves people just two unsatisfactory choices in US.
The example tells about inability to reach conclusion in debates, but explicitly assumes good faith, you probably skipped the final part. And in real life as well, not a lot of people would be inclined to choose civil war instead of political/cultural one.
*Btw, usually there are more then two sides: it's just stale majoritarian system which leaves people just two unsatisfactory choices in US.