I agree with Lilo. In the fishtank scene, Vito asks, "Do you trust my judgment?" "Always, Godfather," says Clem, with feeling. "Yess-ss," hisses Tess. If I remember correctly, Tess exits hastily, without a handshake. I think he sent out feelers to Barzini soon afterward. I don't think Barz attempted to turn him first, much less at Vito's burial.
Tess shakes Michael's hand, but does so with his LEFT hand. This is telling. Much like when Luca refused to shake hands with Tatt Jr.
I am sure Tess put feelers out right after the meeting, and it was even clear to an ailing Vito that there would be a traitor in the family as soon as he died.
Tess and Barz, who underestimated Michael, figured he'd be easy game once he took over.
