Don’t worry, I have been researching it.
The only places I’ve ever heard that the priest was the minister of Matrimony in the Eastern Churches was here on CAF and on Wikipedia.
So now you’ve heard it here on CAF, on Wikipedia, and from the Catechism of the Catholic Church.
The Eparchy of Newton says this:
Saint Thomas Aquinas (13th Century) said that the vows were the essential matter of the sacrament; this led to them receiving more importance and the other parts of the ceremony being diminished. According to the Canon Law of the Eastern Catholic churches, the essential elements of our ceremony are the consent, the prayers and blessings of the priest, and the marriage crowns.
Since vows are not (usually) included in the Byzantine crowning, how can they be essential to the sacrament? Consent is essential, vows are not.
You seem to only view a source as authoritative if it comes from Rome, and not always then. With this view, Eastern Catholics can never adequately speak for themselves in the absence of affirmation from Rome. The problem is, Rome doesn’t work this way when it comes to Eastern theology. Rome is never going to provide a defense of Eastern theology because it is not her theology. If Rome has an objection to some Eastern theology because it contradicts dogma, Rome will speak.
As Eastern Catholics, we often look to Orthodox sources to learn our own theology. English language sources are limited. I could direct you to any number of Orthodox sources, but you would reject them. It seems that nothing will satisfy you in this. I understand that it is an apparent contradiction and quite puzzling. I have asked my own priest, and others about this contradiction and how to make sense of it. I have not received a completely satisfactory answer, but I am able to trust those who have more knowledge, more understanding and more authority than I have. Every source of authority that I have consulted, from my pastor, to an FSSP priest, to my own bishop, has confirmed that the priest is the minister of the sacrament in the East. Perhaps my understanding will grow someday. If not, I can still accept this teaching because I accept the knowledge and authority of those who teach it. Of course, this is not enough for you and I understand this. I do wonder what is enough for you, though?