shrug
For various secular academics and myself, it is not a religious issue. Ergo I have no emotional desire to see myself proven right.
The current academic environment on the matter is the the Pentarchy had rule of the greater church (though including small, insignificant Jerusalem in it really was a primacy of honor).
As these cities fell to Islamic lords, the enormous Christian populations they controlled contracted, leaving Rome the only honorific of the five to remain under Christian temporal control.
Please note that the Pentarchy wasn’t really a recognized institution. It’s mostly a word used to describe the higher Christian ecclesiastical status quo until the 7th century which saw the fall of possibly the largest at the time; Alexandria.
Until the rise of Islam, Christendom was divided fairly evenly into thirds - Latins with their eyes set on Rome, Greeks with their eyes set on Constantinople and Eastern Christians with their eyes set on Alexandria (and to some extent, Ctesiphon).
After the events in the seventh century, the latter two were Christian realms were believers were steadily culled or converted.
This simply left Rome as the effectively unchallenged Christian center by the concluding 13th century - what scholars describe, as no coincidence, the golden age of Roman Catholicism.
With every contraction of Eastern and Byzantine Christian influence, the remainder of the Christian world was more and more Latin. This, according to seculars on the matter, largely coincided with developments in papal power. The establishment of Papal States. The crowning of a Holy Roman Emperor, the mingling of secular and religious power structures. So on.
There’s no question that papal power grew in time unless you have a religious reason to say otherwise. And to that, again;
shrug