Now that we know what a 'normal' post-Ressurrection Appearance is supposed to look like, we can note what is special about John 21. And it certainly is!
What is being communicated in 21 is the relative importance of two respective leaders: Peter, and the Beloved Disciple.
Chapter 21 presumes that we know the whole text of John (1-20) and its characters, specifically, Peter and the Beloved Disciple.
Peter:
The synoptics state Peter was chosen first and made 'the Rock' of the church. But that's not how John tells it!
(1) In John, Peter is called 2nd,after Andrew, contradicting the synoptic tradition. Peter isn't even called by Jesus, but by Andrew. So John begins, and continues:
(2) The synoptics have Peter give the 'Great Confession', that Jesus is the Christ (Mk 8:29 etc) and Peter is even made charismatic leader for his insight. John's version is clearly weaker: The occasion is the abandonment of Jesus by many disciples, not their acknowledgement of Him as a prophet. Instead of being asked what they think of Jesus, Jesus asks, "Will you also go away?" (6:67) Peter timidly says, "To whom shall we go?" (6:68), and offers the ambiguous "Holy one of God" rather than the explicit "the Christ!" (6:69). This doesn't seem much stronger than Nicodemus' respectful comments.
Peter protests the footwashing, and Jesus responds condescendingly, "You don't know what I am doing now" (13:7). And in fact Peter was NOT ready for sacrifice, bold following (as the beloved disciple did) or fearless confession.
Peter also lacks special insider information, such as the i.d. of the traitor, and must depend upon the beloved disciple, who is truly an intimate insider with Jesus.
Beyond the first 'naming' of Peter as 'Kaifa' (Petros), John's gospel without chapter 21 paints a consistently poor picture of Peter.
Beloved Disciple:
In contrast, the Beloved Disciple is always in the know, bold and fearless in crisis, and an intimate companion to Jesus. Although the beloved disciple doesn't appear overtly until chapter 13, he is always favourably portrayed.
He has the most important secret knowledge (who the traitor is), he boldly follows Jesus right into the High Priest's house, and stands at the cross, unafraid to be publicly associated with the condemned Jesus. Jesus declares him 'brother' and bonds him to His own mother.
On Easter morning the beloved disciple is still upstaging Peter: he runs faster, arrives first, sees and believes (20:8). These actions and insights are never granted to Peter.
The beloved disciple, not Peter, remains faithful, is at the cross, sees and believes again.
Are Peter and the Beloved Disciple rivals? certainly they are deliberately contrasted in John.
John 21:
In John chapter 21 they are compared and contrasted once again!
But this time on every point, Peter is uplifted and restored (or elevated) to the position he has in the synoptics and the early church. John 21 then, corrects the Gospel, and brings it in line with the Apostolic churches, especially Rome, reflecting a time when Peter was acknowledged as a preacher and founder, and perhaps also martyr.
In John 21, the Risen Jesus then restores (or raises) Peter to the position of Shepherd of the Flock - a bold apologetic move, but one reflecting later times (post-Acts, post-Paul).
The Kicker:
But this obvious circumstance just as loudly asserts something about the REST of John: John chapters 1-20 reflects an EARLIER period before Peter's post-Ressurrection experience, before his rise to the leadership of the church in Rome, and before there was any knowledge of his martyrdom and subsequent status and veneration.
Which is to say that John (without the last chapter) is OLD. Probably older than Luke-Acts, likely older than Matthew, and possibly as old as Mark.
And in fact, the evidence all points to John's seeming ignorance and independance of the synoptics as most easily explained by the fact he wrote his gospel first, from first-hand accounts of living eye-witnesses. That is precisely how sections like 6:66-71 read. They are primitive accounts that have escaped the influence of later church dogma concerned with establishing apostolic authority, as the synoptics obviously contain.