The first thought I have on capital punishment is: Why is this an issue?
The numbers involved in capital punishment make this a small change issue: These days less than a hundred per year are publicly executed in the US (and less than 3,000 worldwide)(Contemporary Moral Issues, James E. White, page 195). Compare this to the death count of other activities in which the US government intervenes, such as traffic deaths at 40,000/year, airline fatalities at 150/year, and a guess of several thousands/year for drug-related violence (various Wikipedia articles).
So, again, why is this an issue?
It's an issue because it's designed to be an issue. Throughout history what is now called capital punishment has been a very public way of killing people. First, let's be clear: The non-euphemism name for capital punishment is public execution. It is a form of community decision-making-and-action that comes to us from prehistory: The community decides in some ritual fashion that a person has done something wrong and they should be killed for their act, and then kills them in some ritual fashion.
Since this is a community choice, once the decision has been made, the community gets to participate in the enaction. The community decision making process and the way of killing both vary enormously from culture to culture, and time to time within a culture. The deciding part can be as swift and private as a empowered government official making a choice, or something more leisurely and public such as some form of trial-and-appeal process that can take decades. The killing part can be very up front and visceral such as partaking in a stoning, or it can be quite subdued such as waiting for a news announcement after the killing happens in a fairly private setting. But, even in modern civilized times, it's a public event and highly ritualized.
Because it's a public event, it's also a form of entertainment -- and it's a form that predates ancient Greek plays and choruses. Like morality plays, part of the purpose of execution is teaching a lesson, and part is making the audience feel better. The feel better part is in many ways similar to the scapegoating done by ancient Jewish communities. It should also be remembered that part of the morality of the event can be bringing shame on to the family of the person executed -- if your father/brother/son is left hanging from the hanging tree, it's a vivid reminder to the community of an important event in your recent family history.
(A side note: The contemporary question in the US of whether or not this version or that of killing a person is cruel and unusual punishment is a smokescreen (which also called hand waving) for the real question: Should the government be engaging in this good-old-time form of public entertainment and education, or not?)
As I have just pointed out, the practice of a community engaging in public executions goes back into prehistory. This implies that this is not a rights issue -- the community has the right based on thousands to millions of years of precedence.
Another question is whether or not public execution is compatible with being civilized. As we have become more civilized, have we outgrown this ancient ritual? Does it no longer provide value to the community? Is there still sufficient entertainment, education, and cathartic value in these rituals to make them worthwhile?
For me, the question of whether-or-not is not the big question. For me, reducing the ritual is the important thing to do. In our time, the ritual surrounding public execution in the US has become immensely time consuming and expensive, and I see the huge increase in ritual as the big problem associated with this issue in our modern circumstance. At times I cynically wonder if there isn't some lawyer special interest group that is promoting capital punishment just because it's so lucrative for the lawyers involved.
In summary, public executions are done for the community, not for the criminal or the victim. The question should not be about which methods are cruel and unusual -- that's a smokescreen question -- and it should not be a question of whether or not the state has a right to conduct these. It does. The questions should be about is sufficient good done for the community by conducting public executions, and in what ways can we simplify the ritual?