As I posted above, just because Barzini wanted hegemony over organized crime in NY doesn't mean he'd get it. By backing Sollozzo's drug operation, Barzini might enhance his revenue and influence. However, by providing Sollozzo with what he wanted, the Corleones would benefit in the same way.
If Tessio and Clemenza were potential usurpers, so were Barzini's capos. Sonny's hottness for the Sollozzo's deal was simple hubris; there's no guarantee that Sonny would have played any part in it and Vito would still remain as the conduit for the family's relationship with political and legal authorities.