They are both fools.
The female is a fool for acting in such a way while he is driving. That's not fair to other drivers and puts other people at risk. It's selfish behavior.
The male is a fool for broadcasting it online. Why? Because the internet never forgets. It's personal family business and it should be kept private. The end. This isn't about being a bigger person.
If he were videotaping it so she could watch herself when she's calmed down and use it as a springboard to talk about appropriate behavior, then great. But the very fact that he posted it so he could get some sort of validation from complete strangers online is foolish. You don't post things online of other people without their consent. I highly doubt she consented to having that posted online.
If he wanted to genuinely demonstrate her behavior, videotaping wasn't really necessary. He could have just recorded the conversation and posted that, and left her (and his) appearance out of it.