Any component of a punishment that is intended to deter is wrong. They hypothetical future offenders supposedly being deterred by the punishment are not parties in the defendant's case. Therefore they have no place in the sentencing.

Death penalty should be: take a life, pay with your life.

There is no need to consider deterrence in any shape or form.

The concept of deterrence being a reason for capital punishment only bolsters anti-capital-punishment arguments.

That's not right. Deterrence establishes a causal link between the punished and the criminals of tomorrow, who consider the likely outcome of their crime before committing it.

Punishments have a side effect of deterring. People don't wish to be punished so they don't transgress. It shouldn't be the focus of punishment. One problem is that the severity of punishment can be varied in order to vary the deterrent effect, whereas the punishment should fit the crime. That a certain fair punishment is not sufficiently deterring others is immaterial to a given defendant.

Fair punishments may deter insufficiently. The deterrent effect involves the probability of being caught and convicted. A law enforcement and criminal justice system that is not effective at catching and convicting can just crank up penalties to keep the deterrent effect high.

Well point #2 is obviously not working, and it makes sense, how many murderers are making rational decisions?

Also, who would be deterred by the risk of execution, but not deterred by the risk of a life sentence?

Claiming harsh sentencing is a deterrent is a convenient cover for the real reason - it's about revenge. Because it very obviously isn't a deterrent, at least, not an effective one.

Unless US states that execute people have dramatically lower rates of murder than states that don't, in which case, I'm wrong and stand corrected.

Guess it depends which of rehabilitation or punishment you value most.