The benefit of the capital punishment is to at least relieve the victim’s family of a horrific psychological burden. Perhaps for others, the death penalty would compound the stress and despair. Whatever works best for the survivors makes sense.
In cases such as these, society owes the option of capital punishment to the victim’s family with a 3 year waiting before a decision could be made, another 2 years before the option is rescinded. If the choice is made for execution, it takes place within two weeks. In default after 5 years, the outcome is determined by the State.
I've read of cases, not just tv dramas, where survivors or next of kin ask that the death penalty not be sought.
I believe their wishes are usually respected.