Sollozzo probably was telling the truth when he said, "The Tattaglias are behind me with all their people...the other families will go along with anything to prevent a war." I think what happened was that the other Dons wouldn't move against Vito on their own. But when Sollozzo and Tattaglia came along and said they was willing to take the risk (and perhaps let the other families in on the drugs trade), the others thought they had nothing to lose.
Stavka, Moe said he could make a deal with Barzini "and still keep my hotel," whereas Michael wanted to buy him out. His statement, "I talk to Barzini" (Moe's emphasis), I think was designed to make Michael believe that he had clout.