While not a proponent of the death penalty myself, I think what you describe as making us feel better is one of the five purposes of punishment, retribution.
Punishment has five recognized purposes: deterrence, incapacitation, rehabilitation, retribution, and restitution.
1.5 The Purposes of Punishment | Criminal Law
So one of the stated purposes of punishment is to make us (society) feel better. For those who favor the death penalty, they feel with anything less after a heinous crime like this, justice is not done. They feel better and safer when justice is served.
I get that. When someone molests and kills a child, part of me wants that person to suffer the same pain he inflicted. That part of my brain tells me it would be fair is that happened to him in prison. If someone in Pakistan threw acid on a young girls face because he felt dishonored, my first instinct would be he should have acid thrown on his face.
Upon much reflection and thought, I think for me the death penalty and eye for and eye punishments appeal to my baser instinct., and that's not where I want to go or where I want society to go. I think as society evolves the death penalty will go away, but that might be tens or hundreds or thousands of years from now.
We, or many of us, in the US view punishments like cutting a hand off for stealing or public lashings as barbaric. We, or many of us, would think of public beheadings or hangings as barbaric. But is a beheading that different than lethal injections?
I think in time we as a society will view all executions as barbaric. I do not know what's right or wrong. I completely get the support many have for the death penalty. For me it doesn't seem right. I am glad no one I prosecuted got the death penalty. I sure did ask for it, got one vote away from it with one jury on a triple murder I tried where a couple, and their daughter were murdered.