Yes, some level of detachment can help when advising strangers and caring too much may make people act in counterproductive ways. That holds true in other contexts too. But, that is not the only thing in play.

Some people find it easy to stay calm and patient, some much harder. And some are able to recognize that "I am pissed now, better not respond immediately". Yet others are able to be polite and nice even while pissed - e.g. have above average good emotional control.

No, I mean I care. I just don't let people make me frustrated. With the exception of people who are manipulative or assholes, I don't make other people responsible for my feelings. I try not too. It's not healthy to be like "I'll be calm if you learn to act in ways to not me angry."