You raise a very interesting point, Trilogy. Barzini's first reaction almost certainly would have been that Tessio was trying to trap him. Tess must have told him a long story about Michael's alleged failures and weaknesses. But, I'm guessing, Tess may have offered an alliance with the Corleone family after he took over--perhaps even to share the family's police/political power and some of their rackets. That would have made Tess, in effect, a subordinate of Barzini.

I think if the murder of Michael (and no doubt Clemenza) had come off, Tessio's days would have been numbered. Barzini would have no further use from him, and nothing is as untrustworthy as a caporegime who betrayed his Don to another Don.