Cobb would have been with his gambling/fixing buddy Tris Speaker a bit earlier than when both played for the A's in late 20s.
Both of them (including Smokey Joe Wood) were accused of betting/fixing a game in late 1919 ( a good year for a fix I guess... ie J.Jackson).
Even with Cleveland (with Jackson, Lajoie, and later Speaker) Cobb would have been the same extremely famous player........... lol, even if he changed teams as much as Gaylord Perry did in his career
I dont believe he could have tarnished his career with anything stupid he did ... he was a mean racist game fixing gambler with no friends... yet is 2nd only to Babe Ruth in name recognition/fame for that Era. All because of his play on the field... like it should be IMO