Robert didn't start the rebellion.
Aerys killed the Starks, and demanded the heads of Ned and Robert Baratheon from Jon arryn. John raised the kids, and refused and called in his army to rebel. Ned rallied the north, Robert rallied his lands, they married into the tullys to cement them on their side. The only reason it was Roberts rebellion was because he was the only of the three with royal blood in him from the targs. Jon arryn started the rebellion, Ned was a much more important military leader, but Robert ended up on the throne. They really could have installed any house lord on the throne, but knew targ supports would fall in line more with Robert than someone else after the war. Outside of killing rhaegar, Robert could have been replaced, Stannis was much more important to the rebellion.
Without Jon arryn I doubt there is a rebellion. Ned and Robert were young and brash. No way that tywin, grey joy, and maybe the tullys would ever through in with them if it wasn't for Jon arryn.