So you need to get emotional to make a rational point?
Nice!
Yup. Sometimes people really don't GET it until you allow yourself to show intense emotion. (But it's being used to make a point, NOT because you're out of control.)
I can't begin to tell you how many times people (especially SF types) tend to not "get" how upset I am about something or how much something means to me unless I actually express my emotions instead of trying to sound level and objective about things.
This is practical use (to some degree) of Fe-style behavior, I think. You access the conventional rules accepted in your culture in order to convey your commitments and intentions and internal state.
Hap, yes, the mediation process is not easy. (Kids give you a real workout... and yes, it's like, "Wait, why am I screaming at you to be nice?!")
The best I have been able to do is maintain a sense of detachment... not as in "not caring" but as in, "I can't change them, I can't fix things, all I can successfully do is try to reframe the situation for them, then let THEM decide how to approach it." If you're in an authority position over them, you can also just break it up if it looks like it's gonna get to bloodshed, but otherwise... And sometimes people also need time to work through their emotions before they're willing to do any lip service, either. So... meh. Take the T route... give some clarity, then get the hell out of dodge.
I think I came up with a quote ten years ago that ran like this: "The only thing worse than two fools trying to destroy each other is the fool standing between them trying to make them make up."