Is every Christian his own pope?

Catholics customarily object that Protestant theology makes every Christian his own pope. They also reprove Protestants for allegedly rejecting the traditional interpretation of Mt 16:18, and substituting a novel interpretation which no one in the early church would recognize.

Origen penned the first major commentary on Matthew. It’s interesting to see how he handles the leading prooftext for papal primacy:

[12:10] And if we too have said like Peter, "Thou art the Christ, the Son of the living God," not as if flesh and blood had revealed it unto us, but by light from the Father in heaven having shone in our heart, we become a Peter, and to us there might be said by the Word, "Thou art Peter," etc. For a rock is every disciple of Christ of whom those drank who drank of the spiritual rock which followed them, and upon every such rock is built every word of the church, add the polity in accordance with it; for in each of the perfect, who have the combination of words and deeds and thoughts which fill up the blessedness, is the church built by God.

[12:11] 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," 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"? 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 thee the keys of the kingdom of heaven," 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 thou shalt bind on earth shall be bound in heaven," etc; but in the Gospel of John the Saviour having given the Holy Spirit unto the disciples by breathing upon them said, "Receive ye the Holy Spirit," etc. Many then will say to the Saviour, "Thou art 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" they may speak through the Spirit of God saying concerning Him, "Lord Jesus," and to Him, "Thou art the Christ, the Son of the living God." 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, 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.

http://www.earlychristianwritings.com/text/origen-matthew.html