The Church is centered around the bishop. Each local church contains the fullness of mysteries of the faith (sacraments) and is every which way sacramentally the Church. There is no higher spiritual authority than a bishop.
Administrativley, the organization of the Church could not remain on the early, primitive, levels. Ecclesial structure assinged bishops to oversee larger territories (archbishops), but they are still bishops in terms of dignity and rank, and equal to other bishops. No bishop can lord over other bishop(s), just as Apostles did not lord over one another. hey can, however, address other bishops pastorally, which is exactly what St. Paul does in his pastoral epistles, or what the Pope/Patriarchs do to the churches in communion with them.
The same is true of Patriarchs, whose dignity (honor) is simply historical and political (associated with imperial capitals, and historical churchesOlder and Newer Rome, Alexandria, Jerusalem, Antioch...or of nations in newer times), but they are still "just" bishops.
The official title of the Pope of Rome was Episcopus Romanus (Bishop of Rome) until the end of the 4th century. As far as the ecclesial structure goes in the first millennium, he was first among others in honor and privilegde but not in jurisdiction and power who presides in charity and the elder brother.
The Eastern Orthodox Church is still ecclasially organized in the same way as the ancient Undivided Church of the first millennium. All decisions are decided by a Synod, and a Patriarach, who presides and covenes councils, casts his vote along with other bishops, but has no jurisdictional power over them. On ly the Church as a whole administatively decides on enforcement and rules by a vote in a Synod of bishops. The Church canons are the "constitution" that safeguards order and discipline within the ecclasial community.
"Yet, in both cases, there lacks any serious proof that anything wirtten in the Bible has any factual merit".
There lacks any serious proof that your history of the Church in the first millenium has any factual merit.