Agree, but just to clarify, my understanding is that if the prosecution decides this will be a death-eligible case, then the sentencing jury will decide whether the defendant's punishment is the death penalty or life without parole. The judge has to abide by the jury's decision. If the jury isn't able to decide, then the judge can impose LWOP, but not the death penalty. (Note, I am not a lawyer, just reading Idaho law and sentencing in capital cases online.)
Also, the defendant, BK in this case, can address the sentencing jury directly on the stand, and without it being subject to cross-examination to present mitigating factors - mitigation against the death penalty. And the sentencing jury may consider BK's testimony in its deliberations.
In addition, families of the victims can make victim impact statements before the sentencing jury, but they should not address whether the appropritate sentence should be LWOP or the death penalty. If family members do address the issue of what they see as the appropriate punishment, then the jury must disregard these statements.
Also, the sentencing jury must decide if the evidence justifies "aggravated" murder in order for the death sentence to be imposed. They can either consider evidence presented during the trial phase of the case, or evidence presented during the sentencing phase. If they don't find that there is evidence for aggravated murder, then no further deliberations are necessary and the judge will impose a sentence of LWOP.
Just my interpretation of the following material that I read online about Idaho judges' instructions to sentencing juries during capital cases. These instructions may have changed over time, but most of the information is likely still relevant.