Stannis didn't discovered Cersei/Jamie's secret when he was on the small counsel. He wouldn't have ran to Dragonstone, he would've told Robert...maybe even murdered them himself. Ned told him about those b*stards, and then at the beginning of season 2 Stannis proclaims that he "won't make the same mistake as Ned Stark" (by only telling Stannis about them) and he sent ravens to all the major houses in the 7 kingdoms.
And Cersei didn't poison John Aryn, although Stannis may believe that and it may have prompted him to leave the council, I can't remember