The issue is that the figure of the Pope is something of an oddity.
The medieval catholic church was an international intelligencia, a combination of Maester and Septon if you will

. IiRC few medieval kings were literate until the late era. This together with church tithes gave the head of the institution a lot of power and prestige. More importantly was his ability to leverage spiritual authority into political influence- Charlemagne being the best example, but many kings were crowned by the Pope- Poland and Hungary IIRC, also William the Bastard also had his conquest legitimized by the Pope. They would play local lords off against one another to prevent any one from dominating Itsly- they did it with the Normans in Sicily and he Germans against the Hohenstaufen's.
Feudal kings rested on two principles- the feudal contract, a set of reciprocal privileges and duties between liege and vassal, and the Divine Right; kings were seen as gods vicars in a sense, appointed to sheperd the flock and maintain the land. Thus all he land of a kingdom was theirs in some sense, entrusted by God to be parceled out among their subjects on lease. Actually ll land was seen as belonging to God- therefore the Pope could grant rulers right to manage that land if the leader failed their holy duty (not paying church taxes, attacking the Papacy, supporting heretics, etc...)
Further the person of the king was invested with spiritual autbority- to strike a monarch, or insult them (calling them stupid, evil etc.) Could be seen as almost blasphemous.
And this is why the Pope was so powerful- he could declare kings excommunicated, making them "fair game" to be defied or deposed by their vassals and absolving said vassals of their obligations of fealty. This is what happened to the emperors during their conflicts- the pope essentially gave the German princes the right to rebel, forcing them to seek absolution from the Pope. Furthermore he could essentially draw up land for food Catholics. This is why the Spanish were so big on conversion- the whole justification for their rule over America was a spiritual mission to save the souls of the native peoples. This is also why you had such things as the Treaty of Tordesillas.
This all changed in the early modern era, when the Habsburg's secured the Imperial title. Now the Catholic church became a symbol of support for Imperial authority rather than a mechanism for opposition.
So the long and the short of it is that the Papacy is a special snowflake. An emperor- German or Greek- could conceivably reassert the primacy of their authority over he Church. A leader could concievsbly declare himself head of the church a LA England and many Protestants. A leader could somehow conquer Rome itself, nipping the Papacy in the bud if it happened early on. But all of these would effectively destroy the Papacy as we know it- at best you might get something like the Patriarch of Constantinople during either the ERE or the Ottomans, with other kingdoms breaking g away under autocephalous churches. But the international character of the Papacy almost precludes a temporal leader absent a massive empife- the closest would be a Caliph but Islam is a very different tradition, with spiritual and temporal power bound up together from its inception. And even there the Caliph wasn't necessarily universal- you had the Ummayad split, and different t sects disputed the caliphate.
I think a more interesting proposal is a pope becoming king- they basically were after the 16th century, ruling much of central Italy. A Papal wank leading to a unification of Italy under the Pope perhaps?