I always assumed that the "I want to thank Don Barzini.." line was Vito subtly indicating that he knew Barzini was involved because Barzini was quick to help arrange a meeting once the Corleones had "lost". But it never did quite make sense that he wouldn't have naturally assumed Barzini from the start given that the Tattaglias were considered a weak family (they were the weakest IIRC) so would have been stupid to start an all-out war with the Corleones unless they had some allies to actually give them aid (as opposed to just siding with them to avoid a war the way Cuneo and Stracci did in the novel).

I get the meeting confirming that it was Barzini, but Barzini should have been the top suspect from the onset of the war. Sollozzo wouldn't have struck without someone backing him, which was Tattaglia as far as manpower, but with how overall weak the Tattaglia family was it should have been obvious they needed someone strong to back THEM up, and the only person capable and ambitious enough to do that was Barzini.