I will give a reasoned argument a shot. I am an American citizen, residing in the state of Texas. I have a right to vote, and a responsibility to serve on a jury in the event that I am summoned to do so. Within the state of Texas, we are all subject to the rule of law. In the event that a person is charged by the people and convicted by a jury within the jurisdiction of the state of Texas, that person will be subject to a penalty proscribed by either the jury or a judge. All penalties have been established by the Texas state legislature, and reviewed by the district, state, and supreme courts. There is an established appeals process, as well as a gubernatorial clemency option. Our citizens have not convicted, sentenced, and executed any individual without due process and proper representation; inclusive of appeals.
And you could have said, "if you don't like it don't move to Texas!"
Even with due process - mistakes do occur. And even due process comes into question in some cases. There are time-limit laws in place in many states where new evidence is not allowed after x amount of time after the trial. Evidence that clearly exonerates the convicted is not allowed to even be introduced to the courts for consideration.
As I've said before, the argument revolves around whether people find it acceptable for innocent people to be put to death for the greater good'.
I don't think it is an unreasoned position to say that, as a society, we are willing to accept this. I happen to strongly disagree with this position.