Can’t you do a rebellion without overthrowing the dynasty? It seems like the obvious choice would be to get rid of Aerys and Rhaegar, who have violated the social contract and proven themselves unfit to rule, and put in a regency for young Aegon. To go to historical examples, this is more or less what the barons did when they overthrew Edward II, and that provided for a lot more stability than the dynastic struggles of the 15th century.