I looked up some Patristic quotes, trying to get a sample of luminaries looked up to by the west as well as the east. Frankly, no where do the fathers I looked up discuss the papal claims.
They do seem to discuss the capacity of Christ's love to forgive St. Peter for his three-fold denial of knowing Christ, and indicating that St. Peter does have a task ahead of him: to serve as a Apostle, but not necessarily the King of Apostles. St. john Chrysostom does mention that St. Peter is the leader of the Apostles, but no where does he mention that this leadership is transmitted to his future 'descendants', and only those decendants in Rome but not Antioch.
Here's what Augustine wrote about these verses in his tracts on Jonh's gospel:http://www.ccel.org/ccel/schaff/npnf107.iii.xciii.html#iii.xciii-p10.1
In fine, before the Lord’s passion, his slavish fear was questioned by a bond-woman; but after the Lord’s resurrection, his free love by the very Lord of freedom: and so on the one occasion he was troubled, on the other tranquillized; there he denied the One he had loved, here he loved the One he had denied.http://www.ccel.org/ccel/schaff/npnf107.iv.viii.html#iv.viii-p19.1
Charity therefore we commend; charity this Epistle commendeth. The Lord, after His resurrection, what question put He to Peter, but, “Lovest thou me?” And it was not enough to ask it once; a second time also He put none other question, a third time also none other. Although when it came to the third time, Peter, as one who knew not what was the drift of this, was grieved because it seemed as if the Lord did not believe him; nevertheless both a first time and a second, and a third He put this question. Thrice fear denied, thrice love confessed. Behold Peter loveth the Lord.
Here's some quotes from St. John Chrysostom:http://www.ccel.org/ccel/schaff/npnf114.iv.xxxv.html#iv.xxxv-p43.3
Yet Peter in many places ÃƒÆ’Ã¢â‚¬Å¡Ãƒâ€š is moved, and speaks more warmly than John. ÃƒÆ’Ã¢â‚¬Å¡Ãƒâ€š And at the end he hears Christ say, “Peter, lovest thou Me more than these?” Now it is clear that he who loved “more than these” was also beloved. But this in his case was shown by loving Jesus, in the case of the other by being beloved by Jesus. http://www.ccel.org/ccel/schaff/npnf114.iv.xc.html#iv.xc-p2.3
[1.] There are indeed many other things which are able to give us boldness towards God, and to show us bright and approved, but that which most of all brings good will from on high, is tender care for our neighbor. Which therefore Christ requireth of Peter. For when their eating was ended, Jesus saith to Simon Peter, “Simon, son of Jonas, lovest thou Me more than these? He saith unto Him, Yea, Lord, Thou knowest that I love Thee.”
“He saith unto him, Feed My sheep.”
And why, having passed by the others, doth He speak with Peter on these matters? He was the chosen one of the Apostles, the mouth of the disciples, the leader of the band; on this account also Paul went up upon a time to enquire of him rather than the others. And at the same time to show him that he must now be of good cheer, since the denial was done away, Jesus putteth into his hands the chief authority among the brethren; and He bringeth not forward the denial, nor reproacheth him with what had taken place, but saith, “If thou lovest Me, preside over thy brethren, and the warm love which thou didst ever manifest, and in which thou didst rejoice, show thou now; and the life which thou saidst thou wouldest lay down for Me, now give for My sheep.”
When then having been asked once and again, he called Him to witness who knoweth the secrets of the heart, ÃƒÆ’Ã¢â‚¬Å¡Ãƒâ€š and then was asked even a third time, ÃƒÆ’Ã¢â‚¬Å¡Ãƒâ€š he was troubled, fearing a repetition of what had happened before, (for then, having been strong in assertion, he was afterwards convicted,) and therefore he again betaketh himself to Him.
What then was it which caused this especial love? To my thinking, it was that the man displayed great gentleness and meekness, for which reason he doth not appear in many places speaking openly. And how great a thing this is, is plain also from the case of Moses. It was this which made him such and so great as he was. There is nothing equal to lowliness of mind. For which cause Jesus with this began the Beatitudes, and when about to lay as it were the foundation and base of a mighty building, He placed first lowliness of mind. Without this a man cannot possibly be saved; though he fast, though he pray, though he give alms, if it be with a proud spirit, these things are abominable, if humility be not there; while if it be, all these things are amiable and lovely, and are done with safety. Let us then be modest, beloved, let us be modest; success is easy, if we be sober-minded. http://www.ccel.org/ccel/schaff/npnf205.titlepage.html
St. Gregory of Nyssa used this passage in his fight against Eunomius' second book; no where does it discuss Peter being made a King and this power being descended throguh his 'line' to other bishops of Rome and not Antioch. I won't quote it here due to its length; you can look it up if you want to.http://www.ccel.org/ccel/schaff/npnf206.vi.ix.i.html
Here St. Jerome in his dialogue against the Pelagians even discusses the nature of the Church and does not discuss the claims put forth by the Latin Church:
18. C. How is it then, Atticus, that the Divine Word urges us to perfection?
A. I have already explained that in proportion to our strength each one, with all his power, must stretch forward, if by any means he may attain to, and apprehend the reward of his high calling. In short Almighty God, to whom, as the Apostle teaches, the Son must in accordance with the dispensation of the Incarnation be subjected, that“God may be all in all,” clearly shows that all things are by no means subject to Himself. Hence the prophet anticipates his own final subjection, saying, “Shall not my soul be subject to God alone? for of Him cometh my salvation.” And because in the body of the Church Christ is the head, and some of the members still resist, the body does not appear to be subject even to the head. For if one member suffer, all the members suffer with it, and the whole body is tortured by the pain in one member. My meaning may be more clearly expressed thus. So long as we have the treasure in earthen vessels, and are clothed with frail flesh, or rather with mortal and corruptible flesh, we think ourselves fortunate if, in single virtues and separate portions of virtue, we are subject to God. But when this mortal shall have put on immortality, and this corruptible shall have put on incorruption, and death shall be swallowed up in the victory of Christ, then will God be all in all: and so there will not be merely wisdom in Solomon, sweetness in David, zeal in Elias and Phinees, faith in Abraham, perfect love in Peter, to whom it was said,“Simon, son of John, lovest thou me?” zeal for preaching in the chosen vessel, and two or three virtues each in others, but God will be wholly in all, and the company of the saints will rejoice in the whole band of virtues, and God will be all in all.