I can still think of situations in which the understandings of you, the individuals arguing, are not all that is at stake. It also seems to me that you're expecting people to aim to come into conflict with each other, one way or another, whereas this would be a system designed to inhibit discourtesy to others.
In the Viking age, duels were often fought not by the offender and the offended, but by selected "champions", who would take up the cause of either party and duel on their behalf. In peace time, I could see this as being a good use to put soldiers to, aside from whatever other help they are to the society. Conversely, "The goal of the honourable duel was often not so much to kill the opponent as to gain "satisfaction", that is, to restore one's honour by demonstrating a willingness to risk one's life for it".
Perhaps it's more a thing for a warlike society. Being an advocate of warlike society, I advocate dueling as one means of righting wrongs, though by far not the only one.