The key to moving away from persudability as prerequisite in discourse is to operate in, and assume in others, good faith. Good faith does not mean that people are convincible, it means that each is arguing with the intent of honestly explaining themselves, not picking and choosing what they say in hopes of winning.
Do not assume that the person who wants to tear down the Confederate statue is really using it as a pretext to some broader political goal. Do not assume that the person who wants to protect the statue is really motivated by racism. Allow the person you are talking with to elucidate his or her ideas based on his or her own understanding, not on yours. Often when we do this blinders fall away, and we see all sides as part of a broader universe of phenomenon.
Conditions are not ripe for this kind of change in the nature of our discourse. It doesn’t help that all of us walk around with encyclopedias in our pockets. At any moment everyone on every side of every issue can summon statistics, opinions, and quotes. These materials too often take the place of actual dialogue. This is not to say that facts don’t matter, but rather that they can stand in the way of expressing our personal ideas to each other. Instead of reaching into our souls for replies, we reach into Google’s library of Babel to defend, not explain, our ideas.