So for those who are interested:
First, I'll say that I do think there is enough Biblical evidence that Peter was a special apostle, but then that must immediately be qualified by the fact that he was, after all, an apostle. Of the twelve, he was one of the inner three (Peter, James, and John) who were permitted to accompany Jesus at the raising of Jairus' daughter (Mark 5:37), the transfiguration (Mark 9:2), and the trial in Gethsemane (Mark 14:33). He is always named first in the lists of apostles, but this says little. There is the famous Matt 16:17-19, with the caveat that Jesus immediately thereafter rebukes him and calls him Satan (Matt 16:23). And in Matt 16:17-19, although Peter is called the rock on which the Lord will build His Church (I'm aware of other interpretations, but the play in Aramaic is pretty indisputably Kefa/kefa), the New Jerusalem (a metaphor for the heavenly kingdom) is said to be built on twelve foundation stones with the names of the twelve apostles (Rev 21:14; cf. Eph 2:20), and the power to bind and to loose is soon after given to the disciples collectively (Matt 18:15-20), and in the later context it clearly has to do with forgiveness of sin. The only thing uniquely given to Peter in this context is the "keys of the Kingdom," which are never defined, and their relationship to the "key of David" mentioned in Rev 3:7 is unclear. (Some people link the keys to Isa 22:20-22, but I see no textual reason to do so.) Nevertheless, Peter is the most mentioned apostle in the Gospels, he is particularly rebuked for the disciples falling asleep in Gethsemane (Mark 14:37), and he is the most vocal apostle in the early chapters of Acts, so some special place for Peter seems warranted.
On the other hand, the apostles dispute who is the greatest of them (Matt 18:1), and Jesus doesn't answer, "Peter." Such a dispute seems difficult if Peter was supposed to hold a higher rank than the apostles. Similarly, despite some attempts to wrestle out of it, at the Council of Jerusalem it is James who gives the decision (Acts 15:19), and in Peter's own writing he emphasizes his equality with the presbyters (1 Peter 5:1-4) and all believers (2 Peter 1:1). Against the modern Roman Catholic understanding of papal primacy, it must be urged that the Pope had no unique title until the fifth century at the earliest (before that, he was most often called "bishop of Rome," and the title "Pope," which comes from the Greek word for "grandfather," was originally used for all bishops, and then for all Patriarchs of the five Patriarchal sees; the Coptic Patriarch of Alexandria still uses the title "Pope" to this day). It is also noteworthy that the Popes couldn't call ecumenical councils before the year 1000.
Instead, even granting that Peter was called to be "the Prince of the Apostles" (which is, after all, just a forced translation from the Latin phrase meaning "the first of the Apostles"), Peter has important teaching on what it means to be a Christian leader: "You know that those who think to rule over the nations lord it over them and their great men exercise authority over them. But it is not this way among you, but whoever wants to be great among you will be your servant, and whoever wants to be first among you will be the slave of all. For even the Son of Man did not come to be served but to serve, and to give His life as a ransom for many." (Mark 10:42-45) This teaching was heard by Peter and echoed by him in his counsel to presbyters (1 Peter 5:3). So, Peter understood that if he was called to be the first of the apostles, that meant he was to "be the slave of all." It is conspicuous that in the traditional Catholic interpretation of John 21:15-17, they take the command of the Lord to Peter, "Feed my sheep," and conclude that all the Lord's sheep are entrusted to Peter, and so Pope Boniface VIII in 1302 concluded that it was necessary for every person's salvation that he be subject to the Pope. But this is to make the Lord's command to Peter into a command to the sheep! The Lord says nothing to the sheep, here or elsewhere, about listening to Peter, but rather commands Peter to feed the sheep. The duty is Peter's to serve the Lord's flock, not the flock's to serve Peter.
The question of whether this primacy was inherited by "the successors of Peter" is a tricky one exegetically, but I'm prepared to grant that on the basis of the universal interpretation of the Fathers. (I can respect, however, those who do not grant this so easily.) But if my interpretation of Peter's primacy is correct, the Popes may be granted a primacy indeed, a primacy to be what Pope St. Gregory the Great declared they were in the 590s: the servant of the servants of Christ. The papacy exists for the sake of the Church, and not the Church for the sake of the papacy. Papal privilege is not a charism of infallibility (as presently claimed), nor a supreme earthly authority (whether ecclesiastical, temporal, or spiritual, as more traditionally claimed by the Popes), nor a primacy of honor (as was charitably conceded by the bishops in the Eastern Roman Empire), but rather the privilege to lead the charge of Christian humility, to make visible Christ's love and grace, to lead the Church's service to God and neighbor, and to be consumed for the benefit of all Christians everywhere.
What do you think: would the ministrations of the servant of the servants of Christ be objectionable to Protestants and Orthodox? (Of course, some of the past Popes' attempts to "help" other Christians are not what I intend, but rather an actual attempt by the Pope to do what is best for other Christian communities.)