I hope to demonstrate how blatantly dishonest the source of those Patristic quotes is (Catholic Answers).
The quote from Pope Gregory the Great was brutally mutilated and presented as follows:
"Your most sweet holiness, [Bishop Eulogius of Alexandria], has spoken much in your letter to me about the chair of Saint Peter, prince of the apostles, saying that he himself now sits on it in the persons of his successors. And indeed I acknowledge myself to be unworthy . . . I gladly accepted all that has been said, in that he has spoken to me about Peter’s chair, who occupies Peter’s chair. And, though special honor to myself in no wise delights me . . . who can be ignorant that holy Church has been made firm in the solidity of the prince of the apostles, who derived his name from the firmness of his mind, so as to be called Peter from petra. And to him it is said by the voice of the Truth, ‘To you I will give the keys of the kingdom of heaven’ [Matt. 16:19]. And again it is said to him, ‘And when you are converted, strengthen your brethren’ [Luke 22:32]. And once more, ‘Simon, son of John, do you love me? Feed my sheep’ [John 21:17]" (Letters 40 [A.D. 597]).
The entire letter reveals something startlingly different however:
TO EULOGIUS, BISHOP.
Gregory to Eulogius, Bishop of Alexandria.
Your most sweet Holiness has spoken much in your letter to me about the chair of Saint Peter, Prince of the apostles, saying that he himself now sits on it in the persons of his successors. And indeed I acknowledge myself to be unworthy, not only in the dignity of such as preside, but even in the number of such as stand. But I gladly accepted all that has been said, in that he has spoken to me about Peter's chair who occupies Peter's chair. And, though special honour to myself in no wise delights me, yet I greatly rejoiced because you, most holy ones, have given to yourselves what you have bestowed upon me. For who can be ignorant that holy Church has been made firm in the solidity of the Prince of the apostles, who derived his name from the firmness of his mind, so as to be called Petrus from petra. And to him it is said by the voice of the Truth, To thee I will give the keys of the kingdom of heaven (Matth. xvi. 19). And again it is said to him, And when thou art converted, strengthen thy brethren (xxii. 32). And once more, Simon, son of Jonas, lovest thou Me? Feed my sheep (Joh. xxi. 17). Wherefore though there are many apostles, yet with regard to the principality itself the See of the Prince of the apostles alone has grown strong in authority, which in three places is the See of one. For he himself exalted the See in which he deigned even to rest and end the present life. He himself adorned the See to which he sent his disciple as evangelist. He himself stablished the See in which, though he was to leave it, he sat for seven years. Since then it is the See of one, and one See, over which by Divine authority three bishops now preside, whatever good I hear of you, this I impute to myself. If you believe anything good of me, impute this to your merits, since we are one in Him Who says, That they all may be one, as Thou, Father, art in me, and I in thee that they also may be one in us (Joh. xvii. 21). Moreover, in paying you the debt of salutation which is due to you, I declare to you that I exult with great joy from knowing that you labour assiduously against the barkings of heretics; and I implore Almighty God that He would aid your Blessedness with His protection, so as through your tongue. to uproot every root of bitterness from the bosom of holy Church, lest it should germinate again to the hindrance of many, and through it many should be defiled. For having received your talent you think on the injunction, Trade till I come (Luke xix. 13). I therefore, though unable to trade at all nevertheless rejoice with you in the gains of your trade, inasmuch as I know this, that if operation does not make me partaker, yet charity does make me a partaker in your labour. For I reckon that the good of a neighbour is common to one that stands idle, if he knows how to rejoice in common in the doings of the other.
Furthermore, I have wished to send you some timber: but your Blessedness has not indicated whether you are in need of it: and we can send some of much larger size, but no ship is sent hither capable of containing it: and I think shame to send the smaller sort. Nevertheless let your Blessedness inform me by letter what I should do.
I have however sent you, as a small blessing from the Church of Saint Peter who loves you, six of the smaller sort of Aquitanian cloaks (pallia), and two napkins (oraria); for, my affection being great, I presume on the acceptableness of even little things. For affection itself has its own worth, and it is quite certain that there will be no offence in what out of love one has presumed to do.
Moreover I have received the blessing of the holy Evangelist Mark, according to the note appended to your letter. But, since I do not drink colatum and viritheum with pleasure, I venture to ask for cognidium, which last year, after a long interval, your Holiness caused to be known in this city. For we here get from the traders the name of cognidium, but not the thing itself. Now I beg that the prayers of your Holiness may support me against all the bitternesses which I suffer in this life, and defend me from them by your intercessions with Almighty God.
Notice what Gregory says: the see of Peter is ONE in THREE places: Rome, Alexandria and Antioch. The edited quote also seems to create the impression in the introduction to his epistle that Pope Gregory is referring to himself as sitting in the Chair of Peter, when in fact he is clearly referring to Eulogius. The same Pope Gregory says in his letter to Anastasius, Bishop of Antioch (Book V, Epistle 39):
"I have introduced in my letter these words drawn from your writings, that your Holiness may know that your own holy Ignatius [Ignatius of Antioch] is also ours. For as we have in common the master, prince of the Apostles, we must neither of us exclusively claim the disciple of this prince of the Apostles."
The same Pope Gregory writes to the Bishop of Alexandria (Book X, Epistle 35):
"Praise and glory be in the heavens to my saintly brother, thanks to whom the voice of Mark is heard from the chair of Peter, whose teaching resounds through the Church as the cymbal in the tabernacle, when he fathoms the mysteries...."
There you have a Pope who, contrary to the claims of Vatican I and the RCC, never believed the See of Peter to be exclusively the Roman Church. Little wonder did he exhoriate anyone who claimed to be "Universal Patriarch", following in the footsteps of St. Cyprian:
"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. 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 from the mind of a Christian that any on should wish to seize for himself that whereby he might seem in the least degree to lessen the honor of his brethren..." (Book V: Epistle XLIII, to Eulogios, Bishop of Alexandria and to Anastasius, Bishop of Antioch)
"Let each one give his opinion without judging any one and without separating from the Communion those who are not of his opinion; for none of us sets himself up as a bishop of bishops, nor compels his brethren to obey him by means of tyrannical terror, every bishop having full liberty and complete power; as he cannot be judged by another, neither can he judge another. Let us al wait the judgement of our Lord Jesus Christ, who alone has the power to appoint us to the government of his Church and to judge our conduct (Council of Carthage, St. Cyprian.)
It's interesting to note that the renowned Roman Catholic scholar Johannes Quasten, in Volume II of his Patrology series, admits:
From these words it is evident that Cyprian does not recognize a primacy of jurisdiction of the bishop of Rome over his colleagues. Nor does he think that Peter was given power over the other apostles because he states: hoc erant utique et ceteri apostoli quod fuit Petrus, pari consortio praediti et honoris et potestatis (De unit. 4). No more did Peter claim it: 'Even Peter, whom the Lord first chose and upon whom He built His Church, when Paul later disputed with him over circumcision, did not claim insolently any prerogative for himself, nor make any arrogant assumptions nor say that he had the primacy and ought to be obeyed' (Epist. 71, 3)."
On the other hand, it is the same Cyprian who gives the highest praise to the church of Rome on account of its importance for ecclesiastical unity and faith, when he complains of heretics ‘who dare to set sail and carry letters from schismatic and blasphemous persons to the see of Peter and the leading church, whence the unity of the priesthood took its rise, not realizing that the Romans, whose faith was proclaimed and praised by the apostle, are men into whose company no perversion of faith can enter’ (Epist. 59, 14). Thus the cathedra Petri is to him the ecclesia principalis and the point of origin of the unitas sacerdotalis. However, even in this letter he makes it quite clear that he does not concede to Rome any higher right to legislate for other sees because he expects her not to interfere in his own diocese ‘since to each separate shepherd has been assigned one portion of the flock to direct and govern and render hereafter an account of his ministry to the Lord’ (Epist- 59, 14)...If he refuses to the bishop of Rome any higher power to maintain by legislation the solidarity of which he is the centre, it must be because he regards the primacy as one of honor and the bishop of Rome as primus inter pares (Johannes Quasten, Patrology (Westminster: Christian Classics, 1983), Volume II, pp. 374-378).