...plus, I think Barzini promised Tess that he'd be the undisputed head of what remained of the Corleone family.

In the fishtank scene, Clemenza says, "Don Corleone, you said the day would come when me an' Tessio could form our own family" [emphasis added]. One family with co-Dons? Never happen. Tess and Clem either would have had to divvy up the Corleone Family, or fought to the death to be the undisputed Don. When Tess went over to Barzini, I'm sure that part of the deal was that Clem would be whacked along with Michael.