By the end of prohibition, the Irish were tasked with exerting ethnic influence in the upper worlds of business and politics. They became acceptable to the mainstream, still to a point, but accepted. The upper world Irish politicians, cops and businessmen were the necessary conduits into providing the underworld's influence and basic ability to operate. The Mafia is Italian in name but the Mob structures it adheres to is based more on the traditional corruption patterns established by Irish-Americans based on cooperation between lawful and unlawful elements.
The Irish had the political connections here in the U.S. before the Italians did but that's because they arrived earlier. Political corruption, in relation to organized crime, wasn't exactly something new back in Italy. And the Irish and Jewish groups faded away precisely because they didn't have the structure the Italians did.
