Public attitudes on the death penalty in the US are rather complicated.
And when you look at the polling/survey data, it's clear that there are people who hold positions that create a bit of a paradox.
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The overwhelming majority in the US acknowledge the risk that an innocent person could be put to death. Nearly two-thirds openly admit that it doesn't deter crime, over half acknowledge that it disproportionally impacts Black people.
Yet, 60% of people support it.
It's an issue that, I'll admit, I've gone back and forth on.
On one hand, the evidence is cut and dry, the government makes mistakes and the judicial/criminal justice system is not perfect.
Since 1973, at least 190 people who had been wrongly convicted and sentenced to death in the U.S. have been exonerated.
For every eight people executed, one person on death row has been exonerated.
To answer your question, that's why it takes so long...if executions were a "speedy" process, at least 190 people would've been wrongfully murdered by the government for something they didn't do.
On the other hand, if there was (what I thought to be) 100% air tight proof that someone murdered one of my family members, I think most honest people would admit that their initial gut reaction would be "fry 'em, and make it slow and painful"...it's just a natural human emotion, so I can understand the survey results.
But with regards to this particular case, I think the irregular nature of the "evidence" and all of the anomalies warrant clemency. Nobody as ever even claimed that this guy was the one who did the killing. A hotel maintenance worker (who was a self admitting drug addict) killed the person with a baseball bat, and then struck a deal with the state (in order to get life without parole instead of execution, themselves) and claimed that the other guy ordered the killing (despite there being no other corroborating evidence to support the maintenance worker's story)
Even if one feels that that a case can be made for the death penalty in cases where there's air tight evidence, said evidence was lacking for this case.
I think some of these unfortunate people are used as pawns in the death penalty debate, where their actual case (and the details) becomes secondary to politicians trying to show the public which side of the debate they're on.