If someone is being a fool, or an idiot, or a moron, or a hypocrite, or a liar, it would help them to tell them that they are being a fool, idiot, moron, hypocrite, or liar. They may not know. And if they don't know, then how will they know unless someone tells them?
It should be noted that this isn't a prohibition on being angry with someone, brother or sister, or otherwise. Otherwise, God (who is shown to be angry with His people Israel on multiple occasions throughout scripture) would be in violation of it.
Rather, it's saying, don't be angry without cause.
Righteous anger is a good thing.
If you observe a Christian is being an idiot, you should call him out on it, and help him not be an idiot.
It has very little to do with anger, but rather, recognizing and promoting a healthy perspective of reality.
Agreed!
You shouldn't just "bow to our interpretation." But if it's correct, then (assuming your position is in conflict with it) you should humbly recognize your error, and correct it. The only way to determine that is to investigate the opposing view, test it, follow it to its logical and rational conclusions, and compare it to what scripture says. If it's in conflict, then reject it, and show your work.
Just like with any other proposition.
As Clete said: By sound reason and logic, and Scripture.