You’ve mentioned that Ned totally outplayed Tywin by sending people with the king’s banner after Gregor. But would Robert have cared? He didn’t punish the Lannister when Jaime attacked his Hand and bestie in the streets. Would he have thought Dondarrion and co. and some smallfolk worth the bother of dealing with Tywin?

Ned wasn’t his Hand at the time when that happened – he had resigned. 

And yes, Robert would have cared. He was never much into governance per se, but ask Balon Greyjoy about how much he cared about challenges to his rule. 

If it was just Dondarrion and some smallfolk, maybe Robert doesn’t care. But Dondarrion carrying Robert’s own banner makes it personal – Tywin is openly attacking him, disrespecting him, challenging him. Robert HAS to respond or it will become public knowledge that anyone can attack the king’s men and get away with it, and at that point you might as well abdicate.