Amateur apologetic use (so-called because of a glaring lack of support among major historians and biblical scholars) of Matt 16:18 as proof of later Roman Catholic claims imports an anachronistic reading into the text that is neither demanded by exegesis nor found anywhere in the first centuries of the Church.
"The promise to Peter from the gospel of Matthew (16:18), 'You are Peter, and on this rock I will build my church,' which is so central for today's bishops of Rome and which now adorns the interior of St. Peter's in gigantic black letters on a gilt background, is not once quoted in full in any of the Christian literature in the first centuries -apart from a text in Tertullian, and this does not quote the passage in connection with Rome but in connection with Peter. Only in the middle of the third century did a bishop of Rome, by the name of Stephen, appeal to the promise to Peter, he did so in a dispute with other churches as to which had the better tradition. However he was no more successful than Bishop Victor had been fifty years previously. Victor attempted to force through in an authoritarian way a uniform date for Easter, without respect for the character and independence of the other churches, and was put in his place by the bishops of the East and West, especially by the highly respected bishop and theologian Irenaeus of Lyons. At the time the rule of one church over the other churches was rejected even in the West." Kung, The Catholic Church: A Short History
, pp. 40-41.
It would almost seem oversimplistic if it were not so often simply assumed to point out that the bishop of Rome is not mentioned in Matt 16. Neither do we find in the earliest fathers the presumption that the bishop of Rome the sole successor to St. Peter, but the presumption that he is not.
St. Cyprian devotes an entire treatise to interpreting Matt 16:17-19 without so much as a single specific mention of the Bishop of Rome; instead he applies the text to the entire episcopate, with every bishop holding the place of Peter in the local church. http://www.newadvent.org/fathers/050701.htm
Regarding Matt 16:18-19, Jaroslav Pelikan writes "As Roman Catholic scholars now concede, the ancient Christian father Cyprian used it to prove the authority of the bishop—not merely of the Roman bishop, but of every bishop" Pelikan, Jaroslav, The Riddle of Roman Catholicism (NY: Abingdon Press), p. 78.
As Fr. John Meyendorff affirms "A very clear patristic tradition sees the succession of Peter in the episcopal ministry. The doctrine of St Cyprian of Carthage on the “See of Peter” being present in every local Church, and not only in Rome, is well-known. It is also found in the East, among people who certainly never read the De unitate ecclesia of Cyprian, but who share its main idea, thus witnessing to it as part of the catholic tradition of the Church. St Gregory of Nyssa, for example, affirms that Christ “through Peter gave to the bishops the keys of the heavenly honors,” and the author of the Areopagitica, when speaking of the “hierarchs” of the Church, refers immediately to the image of St Peter. A careful analysis of ecclesiastical literature both Eastern and Western, of the first millennium, including such documents as the lives of the saint, would certainly show that this tradition was a persistent one; and indeed it belongs to the essence of Christian ecclesiology to consider any local bishop to be the teacher of his flock and therefore to fulfill sacramentally, through apostolic succession, the office of the first true believer, Peter" (John Meyerendorff, The Primacy of Peter
, p. 89).
"Cyprian, along with his synod of North African bishops, left no room for doubt: 'For neither does any of us set himself up as a bishop of bishops, nor by tyrannical terror does any compel his colleague to the necessity of obedience; since every bishop, according to the allowance of his liberty and power, has his own proper right of judgment, and can no more be judged by another than he himself can judge another' (Acts of the Seventh Council of Carthage under Cyprian, The Judgment of Eighty-Seven Bishops on the Baptism of Heretics)" -Laurent Cleenewerke, His Broken Body, p. 80.
Applying typical canons of textual criticism analogically to the history of an idea, the Cyprianic idea about Petrine succession was primitive and persistent, spanning many centuries and attested widely geographically; it is still found represented today in the Orthodox Church. By contrast the understanding of the text often championed by amateur apologists is later, then sporadic, and geographically isolated.
"The “Peter Syndrome” is the automatic (and unjustified) application of anything about Peter to the bishop of Rome exclusively. This is deeply rooted in Roman Catholic consciousness" (Laurent Cleenwerke, op cit, p. 78).
The Gelasian Decretal, which is often mistakenly attributed to Pope Damascus of Rome, identifies the divine primacy continuing in all three Churches- Rome, Alexandria, and Antioch.
"Chrysostom also calls Ignatius of Antioch successor of Peter. There is no doubt that his reference to “Peter and his successors” applies to the bishops everywhere, not to the bishops of Rome exclusively. In fact, there is a real possibility that Chrysostom’s perception of Peter’s role stems from his view of the episcopate (not the other way around)." -Laurent Cleerenwerke, op cit, p. 84.
Pope Gregory the Great (died 604 AD), held all three Patriarchates which existed in his time -Rome, Alexandria and Antioch- all founded by Peter, were equal in power and authority and all possessed the Keys.
St Vincent of Lerins defined true catholic doctrine as marked by universality, antiquity, and consent. The Cyprianic understanding of Peter and the keys fulfills all three criteria; the interpretation argued by amateur RC apologists fulfills none of them.