(Spoilers main) why didn't any kingdom declare independence during Robert's Rebellion

As it is historical feudatories do claim independence during a major civil war or rebellion or uprising. Why didn't the great houses of Westeros think of declaring independence from the Targaryens. The dragons were extinct by that point so it will be easier to face a Targaryen army. The Lannister's and the westerlands would be the best contenders. They had the manpower and the money. And they could have easily invaded the reach as Mace Tyrell and majority of his army was besieging storm's end. Wouldn't Tywin want to be king rather than just a Hand.ok I'm totally stoned and I don't know what I'm typing .it's 3 am and I should probably get some sleep. Bye