Are you willing to claim that the Primacy of the Bishop of Alexandria, and other Eastern Bishops wasn't recognized also?
They had a primacy, but not over the whole Church. Appeals were made to Rome for justice, not to Antioch or Alexandria.
Since the primacy of the Apostolic See has been confirmed by the merits of St. Peter, the prince and the crown of the episcopacy, by the dignity of the City of Rome, and also by the authority of the holy synods, no one should presume to do anything illicit outside the authority of this See. For the peace of the Churches will finally be preserved everywhere when the whole Church is subject to its supreme ruler. (Corpus Juris Civilis, law added by Valentinian III [ruled AD 425-455])
Is that clear enough? That is what the Empire and the Church recognized. That is what is manifestly shown in all the histories.
The primacy of Rome does not consist of appointing Bishops in every See. It consists in having final jurisdictional authority in faith, morals, and discipline over every Church and every Christian.