How could you honestly think it's not though... ?
"Peter... I'm giving YOU the keys..."
If you were to ask me, I myself would ask is it Peter alone who had the keys?
What is the Catholic understanding of other verses such as:
Mathew 18:18 the other Apostles get the same powers.
Still another verse seems to me to make the same implication; of power to all in heaven and on earth.
John 20:23 "If you forgive anyone his sins, they are forgiven; if you do not forgive them, they are not forgiven."
I'm also of the belief that ECFs also said this...
Tertullian
"What, now, (has this to do) with the Church, and) your (church), indeed, Psychic? For, in accordance with the person of Peter, it is to spiritual men that this power will correspondently appertain, either to an apostle or else to a prophet. "
On Modesty. Book VII. Chapter XXI[1]
It is clear others will also posess the keys, those who are 'spiritual men' in accordance, as the person of Peter. It is not Peter's exclusively.
Hilary of Poitiers
“This faith it is which is the foundation of the Church; through this faith the gates of hell cannot prevail against her. This is the faith which has the keys of the kingdom of heaven. Whatsoever this faith shall have loosed or bound on earth shall be loosed or bound in heaven. This faith is the Father's gift by revelation; even the knowledge that we must not imagine a false Christ, a creature made out of nothing, but must confess Him the Son of God, truly possessed of the Divine nature
On the Trinity. Book VI.37[2]
Augustine
“He has given, therefore, the keys to His Church, that whatsoever it should bind on earth might be bound in heaven, and whatsoever it should loose on earth might be, loosed in heaven; that is to say, that whosoever in the Church should not believe that his sins are remitted, they should not be remitted to him; but that whosoever should believe and should repent, and turn from his sins, should be saved by the same faith and repentance on the ground of which he is received into the bosom of the Church. For he who does not believe that his sins can be pardoned, falls into despair, and becomes worse as if no greater good remained for him than to be evil, when he has ceased to have faith in the results of his own repentance.”
On Christian Doctrine Book I.
Chapter 18.17 The Keys Given to the Church.[3]
"...Peter, the first of the apostles, receive the keys of the kingdom of heaven for the binding and loosing of sins; and for the same congregation of saints, in reference to the perfect repose in the bosom of that mysterious life to come did the evangelist John recline on the breast of Christ. For it is not the former alone but the whole Church, that bindeth and looseth sins; nor did the latter alone drink at the fountain of the Lord's breast, to emit again in preaching, of the Word in the beginning, God with God, and those other sublime truths regarding the divinity of Christ, and the Trinity and Unity of the whole Godhead."
On the Gospel of John
Tractate CXXIV.7[4]
"...the keys that were given to the Church,"
A Treatise Concerning the Correction of the Donatists
Chapter 10.45[5]
"How the Church? Why, to her it was said, "To thee I will give the keys of the kingdom of heaven, and whatsoever thou shall loose on earth shall be loosed in heaven, and whatsoever thou shall bind on earth shall be bound in heaven."
Ten Homilies on the First Epistle of John
Homily X.10[6]
Is this what your understanding would be? Or do you know where it might say that Peter alone had the keys and only those that follow him in one particular See that he founded also exclusively have the keys?
However I'm not able to discuss this, so I guess if you want to discuss my reasons (as given) we'd have to adjourn to PM, or a more general thread
Endnotes
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[1]
ANF04. Fathers of the Third Century: Tertullian, Part Fourth; Minucius Felix; Commodian; Origen, Parts First and Second | Christian Classics Ethereal Library
[2]
NPNF2-09. Hilary of Poitiers, John of Damascus | Christian Classics Ethereal Library
[3]
NPNF1-02. St. Augustin's City of God and Christian Doctrine | Christian Classics Ethereal Library
[4]
NPNF1-07. St. Augustin: Homilies on the Gospel of John; Homilies on the First Epistle of John; Soliloquies | Christian Classics Ethereal Library
[5]
NPNF1-04. Augustin: The Writings Against the Manichaeans and Against the Donatists | Christian Classics Ethereal Library
[6]
NPNF1-07. St. Augustin: Homilies on the Gospel of John; Homilies on the First Epistle of John; Soliloquies | Christian Classics Ethereal Library