Winkler describes it a little differently in his book A Life in Movies. He said he had heard about Robert De Niro from De Niro’s early independent films, The Wedding Party and Greetings, and that after Pacino dropped out, “Bob De Niro came in, and we instantly wanted him to play Mario.”
A fun aside: Harvey Keitel referenced The Gang That Couldn’t Shoot Straight while appearing with Pacino and De Niro at the Hollywood premiere of The Irishman. Director Martin Scorsese introduced Keitel, Scorsese, Pacino and the rest of the cast, and then there was confusion about which way they were supposed to exit. “The gang that couldn’t shoot straight!” quipped Keitel.