At the very least, information secrecy. Nobody but the Kages need to know how to kill off a zombie's personality. They're ninjas, right? They should be able to keep secrets.
Conflict will never be eliminated, because conflict is inherent to the world insofar as different people will always be in competition over scarce resources. Conflict can be resolved peacefully or through force. But removing the necessity for food, for example, eliminates a huge source of conflict -- people will actually fight for food, since they normally need it to live. Real wars are over necessities like food and water, or strategic resources to necessary to defend the sustainability of their economies.
By removing these physical needs, the only source of conflicts to remain will be unnecessary luxuries. Even if zombies do fight over them, nobody gets hurt (except a little butthurt for having their luxury item stolen -- I am ignoring scenarios like Shikamaru burying Hidan alive...). Suddenly, instead of war being a terrible burden to the world, it is just a minor inconvenience in an eternal future. Peace will be effectively redefined to include war, since it does no permanent harm, much like how shooting rubber bands at people does not preclude world peace.
Also, the zombies will have an eternity to learn these lessons.
Although this is all purely hypothetical, it does seem like a big WTF that none of the Kage, Orochimaru, etc. never thought of this.