I guess I don't see any inconsistancy between this and the lineage of the papacy of the Catholic Church since Peter, but we already have established differences in our beliefs.
The inconsistancy I see is that the Roman See appears to me to have developed from simply the leadership of the early Christian church in Rome to attempting to assume control over all Christianity. Early Popes who claimed the title Vicar of Peter as opposed to Vicar of Christ didn't seem to be proclaiming ownership and sole leadership of the entire Christian faith as would appear to have developed later in the Church's history. Nor do I see Peter occupying a Papal position in the Bible. All of that appears to me to be a later development.
Firstly, in the early Church, as well as in the Church today, the Pope does not have sole leadership of the entire Christian faith. (remember, the pope is a bishop) The Pope shares his leadership with all the rest of the Catholic bishops around the world. Bishops are responsible for faith, morals and spiritual leadership of their individual dioceses. The pope does not tell each bishop how to run his diocese. What bishops do is submit themselves in obedience to the authority of the pope, promising to take into account the teachings and direction of the pope in their leadership of their dioceses.
Now if we think of the Apostles in the Bible, Christ obviously put Peter in charge of the Apostles. (He tells Peter to "Feed my sheep," Peter directs the Apostles in the Acts after Christ ascends into Heaven) But the rest of the Apostles also made decisions on their own. This parallels almost exactly the relationship betwen the pope and the bishops today. Think of Peter being the Pope, and the Apostles being bishops. Peter was more of an overseer of the Apostles than a sole leader. It is the same today. The pope is an overseer of the entire group of bishops, there to gently guide, hold together, and ensure unity among all the bishops. The pope is not a sole leader of the Catholic Church, but leads the Church in agreement and with the help of all the bishops in the Church. (Think about it: infallibility applies to both the Pope and bishops. If bishops are infallible too in matters of faith and morals besides the pope, then bishops must have leadership and authority too, which argues against the pope have sole leadership in the Church.)
So, the Roman See (the pope) is more like an overseer of the Church, than the sole leadership of the Church. What happened around the time period in question is that the bishop of Constantinople wanted to have the position of "overseer" himself, (he wanted it to be more like he was sole leader of the Church) instead of taking part in the deliberations of all the bishops together under the "recommendations," (notice I said recommendations, not orders) of the pope.
Finally, because the early Church understood the early popes position as "overseer" it used the title "Vicar of Peter" because Peter was the first "overseer." As the Church grew, generations after Christ it made more sense to clarify the title "Vicar of Peter" and change it into the title "Vicar of Christ," to more clearly illustrate the position of "overseer" that the pope occupies. (Christ in a sense is the "overseer" of the Church from Heaven, and the pope, following in his footsteps, is His embodiment of "overseer" on earth, a physical reminder of Christ the "overseer" in Heaven."
In the Church described in the NT, most of the individual churches are those groups of people led by bishops. Those churches in the NT are more like the dioceses of the Catholic Church today, than what we think of as churches today.
I hope you an understand what I am trying to say; I'm doing this in the middle of the night and am so tired it's hard to think clearly.