I think that one aspect of this question that has to be addressed is where an individual holds their beliefs on the nature of the system. This goes back to a Constitutional perception of whether someone believes in Judicial Activism or Judicial Restraint.
If someone is a follower of Judicial Restraint, then they believe that the Constitution should be followed in the strictest of interpretation, without deviating from its original intent. An extension of this belief is that followers of Judicial Restraint inherently believe that the system does not need to be questioned or reworked; the founding fathers "got it right" and to undermine that with tinkering and modification only proves that the founding fathers made an error. Essentially, followers of judicial restraint believe in the strength and authenticity, and almost the infallibility, of the system. This is not to say they are narrow in thought, but rather believe that the judicial system and process does not possess glaring errors that cause it to be questioned.
A follower in the Judicial Activism brand of thought believes that the Constitution is flexible and is supposed to reflect the times. It is not so much as it is " a living and breathing document," but rather that we have to take context and circumstance into question. The world we live in now is vastly different than that of the founding fathers (Remember that the founding fathers did not grant equal rights to women or people of color or young people.) Followers of judicial activism believe that everything- the Constitution, judicial process and procedure- is subject to examining in light of current situations and contexts. It is not saying that judicial activists believe in no structure, but rather they emphasize that interpretation and adherence to structure has to be taken into account with modern contexts and situations.
Understanding both sets of ideas, we can look at the death penalty. The judicial activist would point out that there is disproportion within our justice system and impacting the execution of the death penalty. They would argue that the incarceration rates of people of color and economically disempowered individuals only proves this. They would also argue that the death penalty assumes that the judicial process has been followed perfectly and in an almost sanctimonious form. If there has been one fraction of a percent in error, they would suggest, the death penalty cannot be embraced for it is final in its judgment. The believer in judicial restraint would believe that the death penalty is sound because the system is sound. They would argue that there are enough opportunities given in the process for the defendant to question the evidence presented, and for them to speak. They would also argue the appeals process does allow the defendant an opportunity to receive due process. If, after all steps of procedural and substantive due process, has been followed, for certain crimes, the state has met its burden and can implement the death penalty. Judicial restraint people would argue that the premise that "a mistake could have been made" undermines our way of delivering justice and discredits the Constitution and the institution of law. This might also give a healthy discussion to pros/ cons.