The implication is that somehow being correct in an argument matters more than tone, which is... childish.

If you want to have a productive argument (as opposed to just being an asshole screaming at passerby) you need to convince people that you are worth engaging.

This means showing good faith and not being overly dismissive (i.e. tone).

You think tone matters more then being correct?

If you're talking about purely conveying information, tone is largely immaterial.

If you're talking about convincing people, proper tone is a prerequisite to conveying information.

For most communication, tone starts out being infinitely more important that anything you're saying. Once you reach an inflection point, tone is sufficient and correctness can take priority. The inflection point varies based on your audience.

I take your point. And I guess tone becomes particularly important when you're trying to communicate an unpalatable truth.

If my tone is such that no one engages constructively, then it's not going to be useful whether or not I'm correct.

If my tone is such that many people engage constructively, then it may well be useful whether or not I'm correct, as either people learn from my contribution or people (including me) learn from corrections to my contribution.

It's more than a little context dependent and YMMV, of course.

Most people do, which is why this site will ban a user for being an asshole, but not for being incorrect.

If you want to convince someone of something, absolutely it does.