Umm.. OK... what in the world do you mean by “despite the teachings of the Catholic Church”?
In the scripture you referenced, Jesus was not repeat not making a statement about capital punishment. As He knew the Pharisees had no interest in adultery other than the chance to force Him into making a stand about Mosaic law or affirming their right to use capital punishment while under Roman rule where the Empire reserved that capability to themselves.
He not only thwarted them, but pointed out their hypocrisy in only presenting the woman ‘caught in adultery’. If they had caught her at it, where was the man? Mosaic law would have them both stoned. By saying ‘those without sin cast the first stone’, He in effect challenged them to put up or shut up regarding the use of capital punishment without the Roman empire’s sanction.
Finally, since Jesus was all about repentance and turning from sin, He didn’t condemn the woman after the Pharisees had balked but instead admonished her to ‘go and sin no more’.
Paul’s letter to the Romans chap 13 affirms rightful authority to administer capital punishment because all authority on earth is derived from God.
The current stance of the Catholic church is anti-capital punishment is it not? The question is at what point in history did it change? They encouraged it use against its enemies in the past.