I would say Tony Salerno had a huge amount of power and influence, regardless of what his actual title was.

He has been directly connected to many big rackets, in NY and in other states, including large scale numbers and loan sharking operations, the construction industry and the waterfront. He also had powerful connections in the "legitimate" world through his lawyer Roy Cohn.

Cosa Nostra members from across the USA would come to meet him to discuss business. Now this could be just because he was the designated Genovese family representative and others were the real power behind the scenes, but this kind of nationwide access still makes him very powerful and influential.

I don't know about the other bosses, but Corallo and Salerno seemed like old buddies and Castellano respected him enough to be partners with him in a big concrete business.