God forgives through the priest, he's not just some schlub standing there as a witness. I doubled checked the prayers the priest says after confession, at least in the Antiochian tradition.
You are right. The 12th edition of the Antiochian "Service Book" says,
"Priest: God it was who forgave David through Nathan the Prophet, when he confessed his sins, and Peter weeping bitterly for his denial, and the sinful woman in tears at his feet, and the Publican, and the Prodigal Son: May that same God forgive thee all things, through me a sinner, both in this present world, and in that which is to come, and set thee uncondemmed before his dread Judgement Seat. And now, having no further care for the sins which thou hast declared, depart in peace."
Some relevant quotes:
Matthew 16:19 And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.
John 20:23 Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained.
"Did you commit sin? Enter the Church, repent for your sin, for here is the physician, not the judge. Here one is not investigated; one receives remission of sins." - St John Chrysostomos
"Just as in the Old Testament the priest makes the leper clean or unclean, so in the New Testament the bishop and presbyter binds or looses not those who are innocent or guilty, but by reason of their office, when they have heard various kinds of sins, they know who is to be bound and who loosed." St Jerome