Havana, I must respectfully disagree with you on some of what you wrote.
To my mind, the Sicilian "Mafia" was the strongest of the three groups. And to that point, they were also the better organized and most intelligently architected of these entities.
So in 1931, after the legendary "Castellammrese War" was over, the "Sicilians" still ruled the roost as far as overall power and nationwide structure went.
Hence, although the vast majority of mainland Italians racketeers joined their ranks -- and the name "Cosa Nostra" was formally adapted, the "Sicilians" still ruled the roost, which is clearly evident by the fact that 99% of all recognized "Families" throughout the United States maintained Sicilians as the Family bosses, with some Calabrians and Neapolitans, etc., being "installed" in secondary key positions in certain families as the underboss, consiglieri, and/or capodecina.
That restriction started to ease up years later. But initially --- almost to a Family --- Sicilians headed ALL borgatas with the exception of Chicago under Capone, and one or two other examples in other cities.