History dictates otherwise.

How do you conclude this theory from what is stated by Cyprian?
Peace
Certainly the other Apostles
also were what Peter was, endued with an
equal fellowship both of honour and power; but a commencement is made from unity, that the Church may be set before as one; which one Church, in the Song of Songs, doth the Holy Spirit design and name in the Person of our Lord: My dove, My spotless one, is but one; she is the only one of her mother, elect of her that bare her (Cant. 9:6) (A Library of the Fathers of the Holy Catholic Church (Oxford: Parker, 1844), Cyprian, On The Unity of the Church 3, p. 133).
Our Lord whose precepts and warnings we ought to observe, determining the honour of a Bishop and the ordering of His own Church, speaks in the Gospel and says to Peter, I say unto thee, that thou art Peter, and on 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 on earth shall be bound in heaven. Thence the ordination of Bishops, and the ordering of the Church, runs down along the course of time and line of succession, so that the Church is settled upon her Bishops; and every act of the Church is regulated by these same Prelates (A Library of the Fathers of the Holy Catholic Church (Oxford: Parker, 1844), The Epistles of S. Cyprian, Ep. 33.1).
Tertullian also interpreted Matthew 16:18 to mean that the rock is Peter. But did he interpret it in the sense of being pro-papal?
If, because the Lord has said to Peter, ‘Upon this rock I will build My Church,’ ‘to thee have I given the keys of the heavenly kingdom;’ or, ‘Whatsoever thou shalt have bound or loosed in earth, shall be bound or loosed in the heavens,’ you therefore presume that the power of binding and loosing has derived to you, that is, to every Church akin to Peter, what sort of man are you, subverting and wholly changing the manifest intention of the Lord, conferring (as that intention did) this (gift) personally upon Peter? ‘On thee,’ He says, ‘will I build My church;’ and, ‘I will give thee the keys’…and, ‘Whatsoever thou shalt have loosed or bound’…In (Peter) himself the Church was reared; that is, through (Peter) himself; (Peter) himself essayed the key; you see what key: ‘Men of Israel, let what I say sink into your ears: Jesus the Nazarene, a man destined by God for you,’ and so forth. (Peter) himself, therefore, was the first to unbar, in Christ’s baptism, the entrance to the heavenly kingdom, in which kingdom are ‘loosed’ the sins that were beforetime ‘bound;’ and those which have not been ‘loosed’ are ‘bound,’ in accordance with true salvation…(Alexander Roberts and James Donaldson, The Ante-Nicene Fathers (Grand Rapids: Eerdmans, 1951), Volume IV, Tertullian, On Modesty 21, p. 99).
In the words just preceding this quote Tertullian explicitly denies that this promise can apply to anyone but Peter and therefore he does not in any way see a Petrine primacy in this verse with successors in the bishops of Rome.
Eusebius: ‘And he sent out arrows, and scattered them; he flashed forth lightnings, and routed them. Then the channels of the sea were seen, and the foundations of the world were laid bear, at thy rebuke, O Lord, at the blast of thy nostrils’ (Ps. 18.14)…By ‘the foundations of the world,’ we shall understand the strength of God’s wisdom, by which, first, the order of the universe was established, and then, the world itself was founded—a world which will not be shaken. Yet you will not in any way err from the scope of the truth if you suppose that ‘the world’ is actually the Church of God, and that its ‘foundation’ is in the first place, that unspeakably solid rock on which it is founded, as Scripture says: ‘Upon this rock I will build my Church, and the gates of hell shall not prevail against it’; and elsewhere: ‘The rock, moreover, was Christ.’ For, as the Apostle indicates with these words: ‘No other foundation can anyone lay than that which is laid, which is Christ Jesus.’ Then, too, after the Savior himself, you may rightly judge the foundations of the Church to be the words of the prophets and apostles, in accordance with the statement of the Apostle: ‘Built upon the foundation of the apostles and the prophets, Christ Jesus himself being the cornerstone.’ These foundations of the world have been laid bare because the enemies of God, who once darkened the eyes of our mind, lest we gaze upon divine things, have been routed and put to flight—scattered by the arrows sent from God and put to flight by the rebuke of the Lord and by the blast from his nostrils. As a result, having been saved from these enemies and having received the use of our eyes, we have seen the channels of the sea and have looked upon the foundations of the world. This has happened in our lifetime in many parts of the world (Commentary on the Psalms, M.P.G., Vol. 23, Col. 173, 176).
Augustine: In a passage in this book, I said about the Apostle Peter: ‘On him as on a rock the Church was built’…But I know that very frequently at a later time, I so explained what the Lord said: ‘Thou art Peter, and upon this rock I will build my Church,’ that it be understood as built upon Him whom Peter confessed saying: ‘Thou art the Christ, the Son of the living God,’ and so Peter, called after this rock, represented the person of the Church which is built upon this rock, and has received ‘the keys of the kingdom of heaven.’ For, ‘Thou art Peter’ and not ‘Thou art the rock’ was said to him. But ‘the rock was Christ,’ in confessing whom, as also the whole Church confesses, Simon was called Peter. But let the reader decide which of these two opinions is the more probable (The Fathers of the Church (Washington D.C., Catholic University, 1968), Saint Augustine, The Retractations Chapter 20.1).