Keep in mind that the Chicago Outfit under Torrio and Capone was not a Mafia family. The Unione Siciliana was Chicago's Mafia. Torrio and especially Capone ran the Outfit more like a corporation. Non-Italians rose to prominence: Jake Guzik was both Torrio's and Capone's second-in-command and actually ran the outfit, alongside Frank Nitti, when Capone went to prison. Murray Humphreys was a top-ranking member of the Outfit for decades.
As far as I know, Gotti Sr. was Italian on both sides of his family. His wife, Vicki, had a partly Jewish father, but under Jewish law, since her mother was not Jewish, she wasn't counted (or raised) as Jewish. And Mob "rules" seldom apply uniformly--especially if your father is the Don.