It wasn't confirmed that she 'chased him off', its a joke the hunters made and never happened. And Thistleclaw was dangerous, he believed in the dark forest ways and trained his apprentice to be a killer. Don't get me wrong, I think most of Tigerstar's actions were by his own. But Thistleclaw did train his hate of kittypets
If you've read bluestars phrofecy, she's a diva, doesn't care about her sister, and hates thistleclaw just because he's ambitious, but he never killed any cat to get his way, and she chases him out of starclan, and his mate snowfur is like sobbing, and she goes insane in the first series!!
how would Scourge be alive? 030 wait.. would she have stopped Thistleclaw from sending Tigerstar to hurt Scourge-kit and then he wouldn't have revenge and then he wouldn't mess with clans and then he wouldn't be ambitious and then he wouldn't have made bloodclan and then he wouldn't have killed tigerstar? O3O XDD
But thn thistleclaw won't choose snowfu, whitestorm won't be born and bone would be alive and firestar would be in TDF because in the first book firepaw almost killed clawface but whitestorm stopped him.