Great question! I'm not too knowledgeable in Roman sources, but I can speak to what we know and don't know about the early Papacy.

The construct of the Papacy as we know it today is a medieval creation; in the first 200-300 years or so of the Christian Church, there would not have been any Christian who felt their ecclesial polity was led by a bishop in Rome (except for, maybe, Christians in Rome). In fact, the word "pope" was more of a general title, and usually referred to bishops of different cities (for example, there are contemporary documents giving Athanasius and Cyprian the title). So in this sense, 2nd and 3rd century lists of an unbroken line of popes were immediately anachronistic, given that there is no evidence that these Roman "popes" (lower-case p) had universal authority or even influence over the universal Church.

It is safe to say that the church in Rome was episcopalian in polity (a hierarchal leadership structure in Christian ecclesiology) leading up to the establishment of the Papacy as we know it. Whether there was a monarchial succession of Popes, however, is up for much scrutiny. For example, with Constantine’s conversion and the subsequent formation of Biblical canon at Nicaea, the Bishop of Rome’s importance seemed to primed for seal. However, Damasus I didn’t even attend the council at Nicaea, and Constantine moved his capital to Byzantium anyway.

Following the "barbarian" invasions in Ancient Rome the 5th century bishops of Rome were able to more lastingly consolidate power in a culture that sought unity in a time of turmoil and uncertainty. Leo "the Great," who some consider to be the first Pope in the modern sense, was able to spread his theological doctrines across Christendom and assert his influence mostly because of the political turmoil of the time. When Attila invaded Italy in 452, it was Leo who met Attila and, somehow, convinced the Hun not to sack the city (Christian “tradition” speaks of Saints Peter and Paul walking up to the parley with Leo, intimidating Attila). When the Vandals did sack Rome in 455, it was Leo who was able to negotiate a stop to the burning of the city. To make a long story short, Leo used these events and others to propagate himself as a political authority in Rome, and for the purpose of this question he marks the first in the line of historically-documented Popes (capital P) that we have.

Which brings us to Catholic tradition, which attempts to create an unbroken line of Bishops of Rome to give the Pope the apostolic authority that justifies his position in the church today (and the episcopal ecclesiology that the Catholic Church structures itself on). Apostolic authority is essential to the Catholic religion because it gives the Pope authority from Jesus himself, who allegedly installed Peter as the first Bishop of Rome. Contrary to (perhaps) popular belief, the New Testament (or other 1st and 2nd century texts, for that matter) does not explicitly (or, in the mind of many, implicitly) outline this installment. Most sources indicate that Peter visited Rome, but whether he led the church there is matter for much debate (in fact, if his ministry mirrored his peers and we consider the political situation in Rome at the time, he probably did not “settle down” to lead the church in Rome as its bishop). Still, as it is reflected in the Second Vatican Council, it is important for Catholics to see Peter as the “First among the Apostles,” as recognized by Jesus himself, so that the Pope reflects this style of leadership as the First Priest of the Church.

Immediately after Peter, our sources for apostolic succession get extremely muddy. Forgetting the 3rd-century texts such as the Liberian Catalogue and Liber Pontificalis—2nd century “lists” of the Bishop of Rome are also contradictory and contain numerous anomalies! Some protestant and secular scholars have proposed that the early Church in Rome operated in some sort of “collegiate episcopacy,” a theory obviously inconsistent with Catholic views on succession. Some lists claim Clement as Peter’s successor, while Eusebius tells us of Clement succeeded Linus and Anencletus after Peter’s death. Regardless, many of these very-early “Bishops of Rome” (to whatever extent they were) were well-known figures in early Christianity. While Eusebius didn’t offer references for much of his early list, it is safe to assume that these leaders existed and were influential in the early church in some capacity. Clement, in particular, is famous for the letter commonly attributed to him that failed to make New Testament canon at Nicaea.

The list of Bishops of Rome that we have today, in their varying forms, mostly consists of historical church leaders who were recognized and their feats recorded by (mostly) Christian historians and church leaders of diverse ethnicities. While biographical details are usually hard to come by for many of these figures, their theological, doctrinal, and canonical contributions in the so-called “Apostolic Age” are documented. Whether or not these leaders functioned in any role recognizable as the one that Pope Francis fills is debatable, if not downright unlikely. Thus, there isn’t much reason to believe that these “popes” didn’t exist, and, as you say, held “some position of authority” within the early church.

Hope this answered your question!

Sources:

Eusebius, The History of the Church

Justo Gonzalez, The Story of Christianity

Henry Chadwick, The Early Christian Church

The Oxford Dictionary of the Christian Church

Second Vatican Council