At first, Barzini's goal it seems was to weaken the Corleone's by eliminating Vito. Figured Sonny would be not only an easier to work with partner, but less respected and he'd be able to muscle in on Vito's political connections and chip away at the Corleone's power. He read Sonny correctly as a hothead who'd flush the Corleone's legitimate strength down the drain.

But when it turns to all out war, Sonny is Boss of one Family battling 4, and he has everyone at a stalemate. He's an absolutely effective war leader. Only a matter of time, too, until the Corleone's zero in on Barzini as the puppet master. So, Sonny has to go. With a weakened Vito, a useless Fredo, and the unknown quantity of Michael, the Corleones will be much weakened with Sonny gone.

Barzini takes the next step and tries to have Michael killed as well, figuring that would leave Vito and his inept son Fredo. Fredo is so useless he doesn't need to be taken out; Give Fredo whatever's left of the Corleone's and make him "Boss" while in reality he's Barzini's puppet.

So, Sonny dead, Michael possibly dead, Fredo isn't a threat and he's in Vegas anyway being bitch slapped by Moe Greene. Al Neri hasn't entered the picture yet, there's no Luca Brasi. The Family is effectively Vito and Tom until Michael comes back.

This is the perfect opportunity to take Vito out. You take Vito out and you have his borderline civilian son Michael and Fredo left. The Old Man's political power goes with Vito, and you're dealing with a much weakened Corleone Family and a Michael who is untested, without guidance, and untried really to succeed him.

Vito being alive is the one thing holding the Corleone Family together.

So, why, either in the lead up to the meeting, or during, or after, didn't Barzini attempt once last ditch effort to take out Vito? He clearly drove himself, with seemingly no back up, no protection. Could easily tailed him on the way to the meeting, or had someone in the Corleone operation flip and set Vito up at some point beforehand, or basically did what they did to Sonny on the way back.

This is literally probably the best opportunity Barzini has to finally take Vito out and render the Corleones basically defunct.

So why doesn't he take it?

For all we know, he tried. The conduct of the war is pretty much a mystery, especially from the Five Families side.

Because, ironically, he and the other Dons needed Vito alive--to settle the war and to protect the drugs business.

In the first, misguided, scenario, killing Vito would cause the Corleones to lose half their political protection, meaning that the Corleones' cops and politicos would be up for grabs. Then Sol and Tatt would be able to leverage a greatly weakened Sonny to get access to the remaining cops and politicos. That scenario depended on no war breaking out --Sol convinced himself, Tatt and probably the other Dons of that because Sonny was hot for his deal and Tom knew it was the right thing. But, Vito survived and a long, costly war ensued. With bullets flying, probably most of Vito's police/political allies stuck with him because they didn't know which side to take. By the time of the Commission meeting, the war was at stalemate, business was suffering--and there was still no deal on drugs. With Sonny dead, Fredo useless and Michael an unknown quantity, Vito was the only one left who could negotiate an end to the war and provide the police/political protection needed for them to make millions in drug trafficking.