What is the wisdom behind advocating for forgiveness and pardon, despite the belief in the necessity of punishing the wrongdoer, and does forgiveness bring about Allah's forgiveness?

What you advised your friend is correct. This is the station of virtue, which none attain save the patient, as evidenced by the noble verse: "Nor can goodness and evil be equal. Repel (evil) with that which is better: then verily he, between whom and thee there was hatred, (will become) as though he were a close protector friend! But none is granted it except those who are patient, and none is granted it except the owner of great fortune." This is among the characteristics of those who suppress their anger and pardon people, for whom Allah has prepared Paradise. This is unless the wrongdoer has committed one of Allah's prescribed penalties and the matter has reached the authorities, in which case pardon is not permissible.