I don't remember reading anything that suggests that Piccirilli was close to De Vito, but maybe I'm wrong....

Keep in mind that Nick Sr. was a Bonanno made member. Anyone who wanted to kill him needed permission from the Bonanno family, and here is where Salvatore Montagna comes in. Montagna visited Rizzuto not long before he was killed and asked him to step down. Nick refused and was killed not long after. Paolo Violi's sons could be involved as well as Giuseppe De Vito, but it is more than likely that Montagna gave the go ahead.