Your personal interpretation notwithstanding, the church does not use this example, or the example of Cain, as objections to the use of capital punishment. Rather, she refers to more explicit statements, such as Gn 9:5-6, and Rm 13:1-4. That’s why she has always held that the legitimacy of capital punishment is Scriptural.
If you think Gen. 9:6 is the equivalent of “eye for an eye, and a tooth for a tooth…”, remember Jesus says: “You have heard that it hath been said, An eye for an eye, and a tooth for a tooth. But I say to you not to resist evil: but if one strike thee on thy right cheek, turn to him also the other:” (Mat. 5:38-39)
If you read verses Rom. 13: 9-10, St. Paul specifically cites several of God’s commandments, including “Thou shalt not kill”, and says it and any other commandment is comprised in this word: “Thou shalt love thy neighbor as thyself”, and that one who loves has fulfilled the Law, and thus does not fall into the concupiscence of the flesh, that of the mind, and that of the spirit, and remains in the light in God. He identifies himself with Christ, and he will share in His Kingdom.
In Rom. 12:19, we read “Revenge is mine, I will repay, saith the Lord.”
There’s human laws and justice and God’s Laws and justice. And, Jesus calls us to be perfect as the Father in Heaven is perfect, which is possible here on Earth as Jesus is proof of that. We should follow His example, and the scene of Him and the adulterous woman is one of the many examples He gave. Jesus said those who were without sin should throw the stones. And, no one struck her, because no one was without sin. So, He confirmed the law that inflicts lapidation on adulterers, but He also saved the woman because not one lapidator could be found. He could have killed her, and it would have been justice, but it would not have been mercy. He gave that soul time and possibility to arriving at repentance and holiness, if she wished to reach them. Those in a position of power should follow the same example: love and mercy.
Finally, Jesus warns one will not only be judged for killing their neighbor, but even for getting angry with them (Mat. 5:22).