Yes, that is what St. Paul is talking about in 2 Corinthians 5:15-20. Where do you see my exegesis differ from the inspired text?
The ministry of confession is what Paul calls reconciliation?
So does everyone. "Confession" is a colloquial term for the Sacrament of Reconciliation referring to the whole by its part, where the penitent confesses (tells about) his sins. Other parts are absolution, petition for mercy, and doing penance.
There is no "ministry of confession" there. It is God who is reconciling man to Himself, not man reconciling himself to God by confession.