How to discuss topics that are emotionally loaded?

That's a good point, that the upset person in the conversation might be prone to be taken less seriously, even by themselves, especially if their reasons are hard-to-describe, but not necessarily wrong. 

Looking back at theses situations through this lens, I actually think at one point I didn't take myself seriously enough.

If my reasons are fuzzy, and I'm upset, it is tempting to conclude that I'm just being silly. A better framing is to view negative emotions as a kind of pointer, that says "Hey, in this topic there is still some unresolved issue. There may actually be a good reason why I have this emotion. Let's investigate where it comes from."

For the non-offended person, I think it already helps a lot to have the possibility in the back of you mind, that a topic may be emotional. For example, many people aren't aware that privacy is a topic that can be emotional for people.