I've always been skeptical about he reasoning advanced for the move against Vito. I think it comes down to Barzini coveting Vito's hegemony over the NY underworld and using Vito's No as a pretext to back Sollozzo's move against Vito. But, the drug business was going to continue with or without Vito and Barzini was going to reap its rewards either way. If Barzini could get rid of Vito in the meantime, so much the better for Barzini.
The immediate justification seems to have been that Sollozzo wanted access to the politicians and judges that Vito had in his pockets. But, as Tom points out, if Vito died, the Corleone family would lose most of its connections, and their value as partners in the drug trade would be greatly reduced. So yes, I think Barzini was simply using Sollozzo and the Tattaglias as a means to gain ascendancy over the Corleones. I think Vito figured it out pretty fast at the meeting with the other families; the way Barzini seemed to lord his newfound power over the other Families, believing that Sonny's war had heavily wounded the Corleones, and left Vito with little choice but to capitulate.
