Bob ... take your time. This is a discussion, not a race.
The three times that Jesus asked Peter if he loved Him was to show publicly that Peter had repented of his denials.
Perhaps. On the surface of things, that is the first impression one takes away.
Peter had earned the distrust of his fellow disciples on his abilities and apostleship and this acknowledgment that Christ accepted his word and worth was to re-instate Peter as a fellow apostle.
Hmmmm .... distrust of his fellow disciples? Is that a gut feeling or one you have drawn from a specific passage in Scripture? According to Acts 1,2,3,4,5,8,15, no one questions Peter's authority to speak for the Church, declare anathemas, and resolve doctrinal debates. Peter is the rock on which the Church is built who feeds Jesus’ sheep and whose faith will not fail.
This was not a position of superiority but one of equality.
That sounds like a conclusion you have drawn from YOPIOS (your own personal interpretation of Scripture). I believe I addressed this in my post #73.
One compelling biblical fact that points clearly to Simon Peter’s primacy among the 12 Apostles and his importance and centrality to the drama of Christ’s earthly ministry, is that he is mentioned by name (e.g. Simon, Peter, Cephas, Kephas, etc.) 195 times in the course of the New Testament. The next most often-mentioned Apostle is St. John, who is mentioned a mere 29 times. After John, in descending order, the frequency of the other Apostles being mentioned by name trails off rapidly.When the names of all the Apostles are listed, Peter is always first. Judas Iscariot, the Lord’s traitor, is always listed last (cf. Matt. 10:2-5; Mark 3:16-19; Luke 6:14-17; and Acts 1:13). Sometimes Scripture speaks simply of “Simon Peter and the rest of the Apostles” or “Peter and his companions” (cf. Luke 9:32; Mark 16:7; Acts 2:37), showing that he had a special role that represented the entire apostolic college. Often, Scripture shows Simon Peter as spokesman for the entire apostolic college, as if he were the voice of the Church (cf. Mat. 18:21; Mark 8:29; Luke 8:45; Luke 12:41; John 6:68-69).
The Primacy of Peter
You are neglecting to factor in Luke 22:32 where Jesus prays for Peter, that his faith may not fail. Jesus' prayer for Peter's faith is perfectly efficacious, and this allows Peter to teach the faith without error (which means infallibly).
Take a closer look at Matt. 16:18 where Jesus promises the gates of Hades would never prevail against the Church. This requires that the Church teach infallibly. If the Church did not have the gift of infallibility, the gates of Hades and error would prevail. And also Matt. 16:19 - for Jesus to give Peter and the apostles, mere human beings, the authority to bind in heaven what they bound on earth requires infallibility. This is a gift of the Holy Spirit and has nothing to do with the holiness of the person receiving the gift.
The context of the situation. One disciple had committed suicide and another had denied Jesus at a critical time, not once but three times!! And it was foretold him. It would be reasonable to assume Peter had self doubts and felt depression as well as just wanting to give up. But his actions were not unforgivable. And Jesus did forgive him and reinstate him in front of the others.