The Corleones are in open warfare against all the other families. So I think Vito's comment about it being "Barzini all along" meant that Barzini was the driving force behind both Sollozzo and Tattaglia as well as being the boss with the muscle and smarts to take on Sonny and win.

Barzini (and Sollozzo) were the up and coming modern mafiosi.
As other posters mentioned elsewhere, probably Barzini hoped that Vito would see Tattaglia's involvement with Sollozzo and go for the deal, not looking any deeper than that.

"When the snows fall and the white winds blow, the lone wolf dies but the pack survives."
Winter is Coming

Now this is the Law of the Jungle—as old and as true as the sky; And the wolf that shall keep it may prosper, but the wolf that shall break it must die.
As the creeper that girdles the tree-trunk, the Law runneth forward and back; For the strength of the Pack is the Wolf, and the strength of the Wolf is the Pack.