I’m sorry, but the 5th Amendment argument is rock solid. The amendment effectively states that capital punishment is permitted if certain requirements are met. While that is restrictive, it is also empowering. For example, consider a law that says you can’t have cocaine unless prescribed by a physician. That too is restrictive, but using your arguments it is a complete ban on cocaine even if one has a prescription. Such a law makes it clear that one can have cocaine if one has the physician’s prescription.
I should add that the 5th and 14th amendments also state that life, liberty and property may not be denied without due process of law. It follows that those things may be denied with due process of law. Within that list we certainly deny liberty with due process by jailing people, and deny property with due process by fining people. To remove the word “life” from that list of things that can be denied with due process, without removing the other two, would be just arbitrary results oriented jurisprudence indefensible by either the text or structure of the Constitution.
All of this is true even accepting your claims that the death penalty violates international norms, or is immoral and ineffective. It is possible for the death penalty to flunk all those tests and still pass the Constitutional test.
And the Constitutional test is the only test a judge is empowered to apply.