> It takes a level of emotional intelligence to communicate in a way which gets to the core of the issue while not making the other person feel bad about it.

Hmm, seems more like you just have to assume that people aren't stupid, and there's a reason they believe the things they do. Until you understand that reason, you just have to keep saying you're confused and ask questions to further clarify.

Once the reason(s) is highlighted, sometimes it's well justified, sometimes it isn't, but at least the core issue can now be addressed.