Human beings are naturally violent, even those who have a veneer of civilisation .This is evidenced by the number of people who believe they could easily kill (with no remorse) anyone who hurt a member of their family, particularly a child. Our court and justice systems are the way that as a society we have chosen to manage that natural inclination .
Historically in some societies (for example Viking) killing someone was considered OK for almost any reason. What was not OK was for it to be done in secret. Openly challenging someone for something (e.g being very irritating) was fine and if you were stronger then then and managed to kill them, there was no punishment. This is because bravery and strength was much more valued than kindness and fairness.
In our modern, westernised society we have chosen other values to live by because if you only value strength and bravery without kindness and fairness then society becomes very very violent.
Of course I would want to kill anyone who hurt/killed my DS (and probably could do it reasonably easily and without much remorse).
It is also likely that I would just accept the first person who looked like they might have done it to hate/kill without really trying to prove it, because I would want justice/revenge so strongly that making sure I had the right person would be secondary to that.
I would probably convince myself it was the right person no matter what as the desire for revenge and the hurt would be so strong that I would be unlikely to be able to see clearly or rationally.
It is because of all this, that I am glad I don't have that possibility and that punishment is handed out in a more impartial way. I also do not want any innocent people to die. This is not a very rare occurrence on death row, it is fact quite common.
I do not agree with the death penalty because I do not agree intellectually with killing people unless there is impartial cast iron clad proof that they have done something terrible and that they are absolutely without hope of remorse or redemption. This level of proof is in fact very very rare. Much more rare than the number of people on death row would suggest
One of the prices of having a society that values kindness and fairness (and not just bravery and strength) is that we may have a few irredeemably bad people who do terrible things who are not killed for that. For me that is an acceptable price to pay for a safer society overall. And societies without the death penalty are safer overall, despite the argument that it is a deterrent, which it clearly is not.
I am not saying that societies are safer because they don't have the death penalty but that they are societies that have fairer systems of justice and less violence generally and part of that ethos is not having the death penalty.