This is the exact reverse of what happened. Before R Yosef and R Goren, the Chief Rabbinate was for life. Then the law was changed to make it a five year term, and R Nissim ZTzL was pushed out to be replaced by the “young blood” R Yosef. (R Unterman ZTzL was already niftar). Then at the new Chief Rabbis’ demand the term was changed to 10 years. When the 10 years were up they wanted to extend it again, but Moshe Nissim was the Attorney General and he fought against it on the grounds that if the job should be for life then why was his father pushed out? So R Eliyahu was chosen to replace R Yosef, and R Yosef never forgave him or Moshe Nissim. When Shas started, the first thing the new MKs did was to destroy Mr Nissim’s political career, in revenge for his refusal to extend R Yosef’s term.