No one is thinking about you as much as you are. At the end of the day, people are reacting to their own feelings, values, and ideas as much or more as they are to anything you said. Taking yourself out of the debate - no matter how personal it feels - is always a good first step.
I had to read this twice. But just to confirm what you mean: We should take our personal feelings out of a debate -- but surely we must still remain in the debate?
Instead, if your goal is persuasion, you should remember that, regardless of how we want the world to work, persuasion in practice is as emotional as it is intellectual.
The thing that never happens when real ideas are at stake is that someone manages to deliver an argument so devastating that the other side reconsiders. I've never seen it happen. What I have seen all the time is gradually shifting someone's thinking until they eventually come around.