What’s with Tywin driving off the Black Ears after the Blackwater? Not only does it belie the claim that “Lannisters pay their debts” but it creates a marauding band with a grudge in the crown’s destabilised lands.

He would argue the Lannisters had paid their debts, that he’d given them arms and armor in exchange for military service and was done.

But as for why he’s doing it, it’s primarily because he doesn’t want Tyrion to have his own military force inside the capital.