As a matter of fact, here are some quotes that prove otherwise:
Pope Gregory the Great (590604 A.D)
"Universal Pontiff" a profane title
For, as your venerable Holiness knows, this name of Universality was offered by the holy synod of Chalcedon to the pontiff of the Apostolic See which by the providence of God I serve(2). But no one of my predecessors has ever consented to use this so profane a title; since, forsooth, if one Patriarch is called Universal, the name of Patriarch in the case of the rest is derogated. But far be this, far be it from the mind of a Christian, that any one should wish to seize for himself that whereby he might seem in the least degree to lessen the honour of his brethren. While, then, we are unwilling to receive this honour when offered to us, think how disgraceful it is for any one to have wished to usurp it to himself perforce.
Wherefore let not your Holiness in your epistles ever call any one Universal, lest you detract from the honour due to yourself in offering to another what is not due.
Pope Gregory the Great criticizes use of the title "Universal Bishop"
and denies Peter was a "Universal Apostle".
For to all who know the Gospel it is apparent that by the Lord's voice the care of the whole Church was committed to the holy Apostle and Prince of all the Apostles, Peter. For to him it is said, Peter, lovest thou Me? Feed My sheep (John xxi. 17). To him it is said, Behold Satan hath desired to sift you as wheat; and I have prayed for thee, Peter, that they faith fail not. And thou, when thou art converted, strengthen thy brethren (Luke xxii. 31). To him it is said, Thou art Peter, and upon this rock I will build My Church, and the gates of hell shall not prevail against it. And I will give unto thee the keys of the kingdom of heaven and whatsoever thou shalt bind an earth shall be bound also in heaven; and whatsoever thou shalt loose on earth shall be loosed also in heaven (Matth. xvi. 18).
Lo, he received the keys of the heavenly kingdom, and power to bind and loose is given him, the care and principality of the whole Church is committed to him, and yet he is not called the universal apostle; while the most holy man, my fellow-priest John, attempts to be called universal bishop. I am compelled to cry out and say, O tempora, O mores!
As you can see, this was written after Chalcedon. Your information is flawed!