It's true that Vito missed seeing Barzini in the Solozzo deal. I've posted earlier that I believe Sol would have gone to Barzini first because he would have figured that Vito, already on top with rackets that were a lot safer than Sol's, would have nothing to gain from the drugs deal--whereas Barzini, younger and more ambitious than the other Dons, would. Barzini would have told Sol: "Your logic is good. But, you still need Corleone's politicians and judges. if Corleone hears that I'm your partner, he'll say no. Go to Tattaglia: he's a pimp, Corleone's not worried about him. But, I'll be your silent partner and I will have your back."

I remember your earlier posts, in addition to the above regarding Barzini being the 'silent partner' because [wanted to dethrone Vito and take over] - among others, extracts: from a couple of your previous posts....Proof that I am a diligent student!

If Sonny had shut up
Barzini was younger than the other Dons, impatient to be Numero Uno, had more to gain from a deal with Sol--and would give him better terms

Tessio/Barzini meeting
Barzini was impatient with being the Number Two Don

Was Barzini's beef personal?
I think Barzini's moves against Vito were "personal" in the sense that he thought of himself as No.1 waiting in the wings, impatient to be the top dog in NYC, younger than the other Dons, whom he regarded as "Moustache Petes" (per the novel). But, I don't think he had a personal grudge against Vito

And ultimately that's why Barzini wasn't as great as Vito. Vito took EVERYTHING personally despite his outward statements that he didn't (take everything personal).