I think you eliminated half the equation. It's not that the jury doesn't agree on the death penalty, it's that the jury doesn't agree on a penalty, one of which is the death penalty. The goal when the death penalty is involved, is to find a jury that agrees on a penalty of life or death -- not one that agrees on death. If you see each phase as distinct trial then it is completely consistent with how our system works. When a jury hangs, the person is retried (if the prosecutor chooses) until a unanimous verdict is reached. And so it is in this phase, the prosecutor can have a new penalty "trial" or can take death off the table and let the judge decide between life and LWOP. I really like the idea that only a jury can decide to put a person to death. I don't think that decision should be in the hands of judge -- especially one who is elected.