Both Tess and Clem wanted to form their own family or families. But since there's no such thing as a co-Donship, they'd either have had to split up the "Olive Oil Busness" between them, or fought it out with only one survivor. Either scenario would have made Barzini stronger.

As we saw, Michael wanted to continue to ,NYC, and Clemenza went along with it. Tessio never would have. The fishtank scene showed that he had no confidence in Michael. That's why he betrayed Michael. Also, the novel tells us that Vito kept Tess on a long leash--unlike Clemenza, he didn't live at The Mall, and Vito liked it that some outside the family thought Tessio was an independent operator--the better to hide Vito's true strength. No way was Tess going to accept being under Michael's long-distance thumb from Neada.