Black Sox had nothing to do with Cobb. And they DID get banned They took money to throw games. Cobb bet and there was a lot more to it.
Though in some way it did. Because the Black Sox weren't actually found out and punished till 1921-22 And some of the players on that Chicago team wound up being witnesses at Ty Cobb's hearing before the Commissioner. But the commissioner could or did not take their testimonty seriously because he had banned them. And in some cases, they were offering their testimony in trade for being let back into the game.
Cobb actually had a full media campaign on his behalf to sway public opinion and it appears to have worked.
But he should have been banned for life at the end of his career as a manager/player for his betting on the game.
Baseball at the Abyss: The Scandals of 1926, Babe Ruth, and the Unlikely Savior Who Rescued a Tarnished Game
Good read