I think it's because, without further context, it's so hard to argue against. Pretty much every person in every culture cares deeply about their children. So if you can successfully hitch your position to that idea, it too becomes hard to argue against.

It's the same with tough on crime. "What, you want criminals to keep getting away with it?!"

> Pretty much every person in every culture cares deeply about their children.

I would substitute "deeply" for "superficially". Like, if my parents found some way to prohibit porn when I was an adolescent, I wouldn't say they cared deeply about me. I would say they were misguided and authoritary. The "care deeply" idea you are putting forward is just trying to distil whatever societal norm currently is into the youngs.