Joe Biden has a brand around being empathetic and a good person, good people don't need to drag families into political debates. His closing argument is overwhelmingly high minded America-above-party stuff. There's no need to go after Trump's kids when if you want to really press his buttons you call out Trump's own failures right to his face.
Trump's neuroses can't deal with his failures, he hides from them and gets angry and outraged by them. He KNOWS he's a fraud and when he's unmasked for it, he goes ballistic. That's probably what happened yesterday in the interview, and that's why Trump only interacts with the Fox News Cinematic Universe on a regular basis, because they won't trigger him.