In any argument at least one and frequently both positions are plain wrong. Both parties think that is true of the other. If that were a reason to drop free speech norms, they would always be dropped.
Civility is an aspirational value, but communities that enforce it invariably will do so in a biased way. Rude behavior is easy to spot in the outgroup and easy to overlook in the ingroup. Offensiveness is nebulously defined- someone might be offended by the word 'asshole', someone else by the banning of vulgarity. How a community defines civility privileges people from backgrounds that define it similarly. Those closest to the issue will have a harder time keeping their emotions in check- eg. a white person might have an easier time arguing dispassionately about racism than a minority.
If you can get around those issues, I think you should. Civil discourse optimizes for a lot of the factors that justify free speech in the first place. Not so with being wrong; a major benefit of free speech is that those who are wrong and participating can be righted, to the benefit of all parties.
"giving the other side a fair chance to make its point" (from very next sentence) doesn't mean they can't blow that chance by being obnoxious, unfair, offensive, or plain wrong. I don't think the OP argues that this "duty to listen" extends forever, that you have to keep listening after you have determined that "the other side" is being an asshole.