Albert Barnes comments
Another interpretation is, that the word “rock” refers to Peter himself. This is the obvious meaning of the passage.
A Catholic commentator writes
Thou art Peter, &c. As St. Peter, by divine revelation, here made a solemn profession of his faith of the divinity of Christ, so in recompense of this faith and profession, our Lord here declares to him the dignity to which he is pleased to raise him: viz. that he, to whom he had already given the name of Peter, signifying a rock, (John 1:42.) should be a rock indeed, of invincible strength, for the support of the building of the church; in which building he should be next to Christ himself, the chief foundation-stone, in quality of chief pastor, ruler, and governor; and should have accordingly all fullness of ecclesiastical power, signified by the keys of the kingdom of heaven.
Upon this rock, &c. The words of Christ to Peter, spoken in the vulgar language of the Jews, which our Lord made use of, were the same as if he had said in English, Thou art a rock, and upon this rock I will build my church. So that, by the plain course of the words, Peter is here declared to be the rock, upon which the church was to be built; Christ himself being both the principal foundation and founder of the same. Where also note, that Christ by building his house, that is, his Church, upon a rock, has thereby secured it against all storms and floods, like the wise builder. (Matthew 7:24, 25.)
You gave a Lutheran's opinion and I give you a Presbyterian and a Catholic opinion.
Another Catholic source says
But if you suppose that upon that one Peter only the whole church is built by God, what would you say about John the son of thunder or each one of the Apostles? Shall we otherwise dare to say, that against Peter in particular the gates of Hades shall not prevail, but that they shall prevail against the other Apostles and the perfect? Does not the saying previously made, "The gates of Hades shall not prevail against it," Matthew 16:18 hold in regard to all and in the case of each of them? And also the saying, "Upon this rock I will build My church"? Matthew 16:18 Are the keys of the kingdom of heaven given by the Lord to Peter only, and will no other of the blessed receive them? But if this promise, "I will give unto you the keys of the kingdom of heaven," Matthew 16:19 be common to the others, how shall not all the things previously spoken of, and the things which are subjoined as having been addressed to Peter, be common to them? For in this place these words seem to be addressed as to Peter only, "Whatsoever you shall bind on earth shall be bound in heaven," Matthew 16:19 etc.; but in the Gospel of John the Saviour having given the Holy Spirit unto the disciples by breathing upon them said, "Receive the Holy Spirit," John 20:22 etc. Many then will say to the Saviour, "You are the Christ, the Son of the living God;" but not all who say this will say it to Him, as not at all having learned it by the revelation of flesh and blood but by the Father in heaven Himself taking away the veil that lay upon their heart, in order that after this "with unveiled face reflecting as a mirror the glory of the Lord" 2 Corinthians 3:18 they may speak through the Spirit of God saying concerning Him, "Lord Jesus," and to Him, "You are the Christ, the Son of the living God." Matthew 16:16 And if any one says this to Him, not by flesh and blood revealing it unto Him but through the Father in heaven, he will obtain the things that were spoken according to the letter of the Gospel to that Peter, but, as the spirit of the Gospel teaches, to every one who becomes such as that Peter was. For all bear the surname of "rock" who are the imitators of Christ, that is, of the spiritual rock which followed those who are being saved, 1 Corinthians 10:4 that they may drink from it the spiritual draught. But these bear the surname of the rock just as Christ does. But also as members of Christ deriving their surname from Him they are called Christians, and from the rock, Peters. And taking occasion from these things you will say that the righteous bear the surname of Christ who is Righteousness, and the wise of Christ who is Wisdom. 1 Corinthians 1:30 And so in regard to all His other names, you will apply them by way of surname to the saints; and to all such the saying of the Saviour might be spoken, "You are Peter," etc., down to the words, "prevail against it." But what is the "it"? Is it the rock upon which Christ builds the church, or is it the church? For the phrase is ambiguous. Or is it as if the rock and the church were one and the same? This I think to be true; for neither against the rock on which Christ builds the church, nor against the church will the gates of Hades prevail; just as the way of a serpent upon a rock, according to what is written in the Proverbs, cannot be found. Now, if the gates of Hades prevail against any one, such an one cannot be a rock upon which Christ builds the church, nor the church built by Jesus upon the rock; for the rock is inaccessible to the serpent, and it is stronger than the gates of Hades which are opposing it, so that because of its strength the gates of Hades do not prevail against it; but the church, as a building of Christ who built His own house wisely upon the rock, Matthew 7:24 is incapable of admitting the gates of Hades which prevail against every man who is outside the rock and the church, but have no power against it.
Previously I quoted from saint Francis De Sales which you may have forgotten.
And earlier in that thread I wrote:
Peter is foundation, not founder, of the whole Church
- foundation but founded on another foundation, which is Christ Jesus, Our Lord ;
- foundation of the Evangelic Church alone,
- foundation subject to succession,
- foundation of the Church militant not of the Church triumphant,
- foundation by participation, ministerial not absolute foundation ;
specifically, administrator and not lord, and in no way the foundation of our faith, hope and charity, nor of the efficacy of the Sacraments,
A difference so great as this makes the one (that is Peter) unable, in comparison, to be called a foundation by the side of the other (that is the Lord Jesus Christ), whilst, however, taken by itself, it can be called a foundation, in order to pay proper regard to the Holy Scriptures.
So, although Christ is the Good Shepherd, he gives us shepherds under himself, between those shepherds and the Lord there is so great a difference that he declares himself to be the only shepherd.