Most people lose the argument in the half second they spend thinking of a comeback instead of listening.

Once your attention moves to what you are going to say, you stop taking in what is being said. You answer a version of their point you assembled in a hurry, they notice you missed it, and now the conversation is about that. The best replies come from hearing the thing accurately, which is impossible while you are drafting.

Try letting people finish all the way, including the pause after. Count one beat before you speak. You will lose a few clever lines and win more of the actual disagreements, which is a trade worth making.