"The Constitution does discuss the death penalty"
Where?
Amendment V
No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a grand jury, except in cases arising in the land or naval forces, or in the militia, when in actual service in time of war or public danger; nor shall any person be subject for the same offense to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty or property, without due process of law; nor shall private property be taken for public use without just compensation.
(boldface added for clarity)
Clearly the death penalty was an expected aspect of "capital trials." Capital punishment means "you lose your head." Therefore, in Amendment 5, "...be subject for the same offense to be twice put in jeopardy of life or limb..." and "nor be deprived of life, liberty, or property" we are clearly discussing capital punishment. To argue that capital punishment is cruel and unusual would be to assume that Amendment 5 did not address the punishments of the day, and therefore, would be revisionist.