if I can add my .02 cents.

In the book THE CORLEONE FAMILY, Barzini, a young Barzini who was not the boss yet, betrays his boss to the Corleone family, in order to avoid a prolonged war. Tessio is the one who kills the boss, Faranzano/Mariposa.

Now I know that this book was written years after all this took place, but it could have been that Barzini actually made his rise to power that way, and is allowing Tessio to do the same.

Tessio would always owe Barzii, like Joe Columbo owed Carlo Gambino. Tessio would need to either bring in Pete or kill him and the rest fall in line....Of course then Barzini would in effect control 3 families, his Tattaglia, and Corleone.