(Elaborating from an earlier answer)

Roman primacy in or over the universal Church have a long, non-linear, history that would require a multi-disciplinary and multi-chronological approach to really contextualize and propose a broader point of view. Still, it might be worthwile to look at how the notion appeared and evolved during the Early Middle-Ages in a period Roman pontifical authority or effective power was far from obvious.

While the Roman bishop claimed a primacy over the other main seats of Christiendom already in the later days of the empire as the successor of St. Peter and the bishop of a symbolical and prestigious capitol in the West (the latter point being notably stressed in Ennodius' Libellus Pro Synodo), nobody really agreed on what this primacy was or how it should be applied : Leo I, with the backing of Valentinian III (who saw there a mean to underscore his own political-religious power), managed to assert these claims in the Council of Chalcedonia in 451 but had to accept that the Patriarchate of Constantinople was formally granted a similar primacy, especially but not necessarily exclusively, in the Eastern empire.

In fact, even in the West, Roman primacy enjoyed only a symbolic authority over the metroplite seats and bishops, reflecting the political realities of the time. Milan or Ravenne's metropolites enjoyed a special relationship with the later Roman emperors or patrices that had their courts in the same cities, whereas the constitution of post-imperial kingdoms in the western provinces had left local clergy under the direct political-religious authority of the new kings, almost always following an Homean credo on top of that (while actual persecution of Nicean clergy was essentally limited to Vandals). Interestingly, the installation of a double Homean hierarchy in western provinces (not as much for proselytism than management) alongside Nicean hierarchy (that still enjoyed more or less good relations with the new masters) gave papacy a certain aura in the West, as the chief seat of orthodoxy. Even there, we're essentially speaking of a symbolic pastoral and legitimizing presence (altough important in expressing one's romanitas), the conversion or re-conversion of western Barbarian to a Nicean credo wouldn't give more actual authority to Rome, kings ideally managing "their" churches as the emperor in Constantinople would do.

The patriarch of the eastern capitol, on the other side, was as close to the imperial court and authority it was possible, admittedly at the cost of his independence, but giving it the means to enforce in reality its claims of primacy over all eastern patriarchal seats of Alexandria, Antioch and Jerusalem. Roman influence there was not neglectible, it was utterly absent, the prestige and authority of its bishop damaged by the collapse of the western Roman seat (a metropolitain seat as Carthage seemingly functioning as a de facto quasi-patriarchate), altough not to the point the claims of having a special relationship with the apostles St. Peter and St. Paul was ever denied.In spite of this difficult situation, Rome wasn't renouncing its claims, nor these considered void in the West or even entierely in the East, as illustrated by the Acacian Schism : the promulgation of the henotikon (act of union, there with myaphisites) without papal approval (or even consideration from Constantinople) gave an opportunity to Orthodox bishops and patriarchs opposed to imperial policy to successfully appeal to Rome whose bishop ended up excommunicating the imperial patriarch, the schism being eventually mended to Roman satisfaction as a firm Nicean emperor took the purple.

This same pattern of Rome appearing in the West, but also the East, as the defender of Orthodoxy against Imperial compromises and religious ingerences; thus increasing its own prestige; would repeat all over the Early Middle-Ages. For the moment being, however, Papacy under the imperial rule of Theoderic wasn't necessarily much more better off : its prestige was restored, insofar that the king recognized it as a religious primate comparable to his own claims of "tutorship" over the former western provinces (and, likewise, factually superficial at best) and that out of a "municipal patriotism" at least, so did senatorial elites, something that the emperors acknowledged as mirroring the state of both "republics" in Rome/Ravenna and Constantinople. But, as in the eastern imperial city, it was at the cost of loosing any real political power to the Amali kings : these did not intervene in pastoral matters, but by influencing as much as possible papal choosings, if not directly appointing pontiffs, made them politically (including in the dealing of religious politics, as during the Acacian Schism) bound. At the least, a sense of "westerness" in Italy, expressed by the permanence of Roman state, senate and religious primacy, would prevail in the region and against Eastern Imperial political expectations in and after the Gothic Wars.

The imperial reconquest would, at first glance, be interpretable as putting back Rome into and under Constantinople's sphere (Justinian proclaming the latter the "chief of all churches"). Popes were replaced andeastern candidates appointed or growing in importance in Roman administration.

But by dealing with with simonic practices and entanglment with local politics in the name of church unity and dignity of the Roman seat, it also connected Rome deeper with the wider "Christian Commonwealth" lead by the Roman Empire in the same time Orthodoxy triumphed in the wake of Imperial and Frankish conquests and the consequential conversion of Gothic and English kings, meaning Rome was even more tied to the theologic debates of the eastern Mediterranean basin and the general religious policies of the Empire (that tended to spill over in the West by other means as well).The emperors couldn't treat the Roman pontiffs as negligible quantity : acknowledged by Justinian as "the firsts among priests", they became an important player in Imperial politics in the West, but also in their religious policies legitimazing their actions, even against Constantinople's patriarchal resistance, Phocas acknowleding popes as "universal bishops" in this context.

Rome, by the late VIth and VIIth centuries, also became a haven for secular or regular clergy fleeing the territories threatened by Avar, Slavic, Lombard and Persian raids or armies; connecting furthermore the papacy with the important christologic debates, eastern patriarchates (especially Jerusalem, with a significant monastic connection between Roman and Palestinian clergy) and more broadly eastern religious politics, radicalizing its position as defender of Nicean and Chalcedonian creeds in the same time these definitely won over the post-imperial West that, while effectively religiously independent when it came to organisation and effective decisions (sometimes bypassing Rome in recieving Imperial influences), all acknowledged its pastoral dominance.

The return of imperial religious policies of compromises (especially the Three Chapters, Monoenergism and Monothelism), ill-recieved by the Orthodox clergy of the eastern provinces as much as they were in the West, provided Rome with opportunities of displaying its claimed role. As Rome was largely under the Imperial thumb (while acquiring a significant autonomy in Latium from ca. 600 onwards), pontiffs were able to sustain imperial repression on religious grounds and while forced to yield, "saved face" insofar both sides were forced to negotiate : emperors couldn't really ignore Rome or just strong-arm the pontiffs they wanted (or even required) a working relationship with eternally, and the Roman bishops were still dependent from imperial legitimacy (the crisis of Three Chapters led to the creation of a band new patriarchate in Aquilea, in rivalry with Rome) but as well protection against Lombards. Eventually, the irrelevance of compromise with miaphysites and monophysites after the Arab conquest paved the way for yet another Imperial-Pontifical normalization.The increased prestige of the pope in both the Latin and imperial Churches made this normalization a rather continuous stream of compromises between the pontiff and the emperor : when Justinian II attempted to impose Greek rites on the Roman church, he eventually backtracked on its assertion and used of a more conciliatory (and efficient) approach as, until the mid-VIIIth century the main focus of both emperor and pontiff was to keep the Christian commonwealth alive.