The ability to seize the throne is nothing. Robert was King. That's why he has the claim of right. He's the rightful heir to his dead brother who was King. That's how primogeniture and inheritance rights works in the Seven Kingdoms, except Dorne, which provides for inheritance to oldest regardless of sex, so women can inherit. Robert was the rightful King because he deposed the sitting dynasty and all of the Lords Paramount swore fealty to him. Starks, Arryns, and Tullys who joined in his rebellion made four Lords Paramount. The Lannisters joined him by sacking KL and slew the royal family except for Crown Prince Rhaegar who Robert killed in combat. The Greyjoys stayed out of the war for the most part and then rebelled and were put down by Robert and his allies. The Tyrells were loyalists and joined the King's Peace later. The Martells were last to join the King's Peace as Rhaegar's wife was Princess Elia of Dorne. They were subdued when Ned rode to the Tower of Joy and engaged in something with the Daynes, the second most prominent family in Dorne.