.
.. as interpreted by you.

Yet the Catholic Church has always taught the permissibility of capital punishment.

The encyclical "Libertas Praestantissimum" written by Pope Leo XIII would help to put things into proper perspective.  A priest had to explain it to me before I could understand how the All-Powerful God can give Authority to a feeble human being and expect the society to accept and respect that Authority.

Since, then, the profession of one religion is necessary in the State, that religion must be professed which alone is true, and which can be recognized without difficulty, especially in Catholic States, because the marks of truth are, as it were, engraven upon it. This religion, therefore, the rulers of the State must preserve and protect, if they would provide — as they should do — with prudence and usefulness for the good of the community. For public authority exists for the welfare of those whom it governs; and, although its proximate end is to lead men to the prosperity found in this life, yet, in so doing, it ought not to diminish, but rather to increase, man’s capability of attaining to the supreme good in which his everlasting happiness consists: which never can be attained if religion be disregarded.