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There is to be one Shepherd here. It was first King David (Ezekiel 34:23), then Jesus (John 10:11), and now is the Pope (John 21:17). This role is tied to the role of King. As David was King (2 Samuel 5:12) and Christ is King (Matthew 27:11), so the Pope is (as the Vicar of Christ) "father of princes and kings".
Historically, the Pope has been the unifier of everyone: emperors, artists, the religious, and so on. Whenever emperors would abandon the Faith, they would be publicly corrected. Whenever artists sought to honor Christ, they would be promoted. And whenever Christians in the East would fall into heresy, the Pope would be the one to bring them in line, as Russian Orthodox theologian Vladimir Solovyov noted.
The Pope has a temporal role. Only the Pope can thoroughly ensure the recognization of Christ as King in society.
Pope Gelasius I wrote to Emperor Anastasius, "There are two powers, august Emperor, by which this world is chiefly ruled, namely, the sacred authority of the priests and the royal power. Of these that of the priests is the more weighty, since they have to render an account for even the kings of men in the divine judgment."
On the correction of earthly monarchs, I often think of the Walk to Canossa, the journey of penance taken by Henry IV, of the Holy Roman Empire, in reparation for his grave sins against the Pope. This was done only out of selfishness on Henry's part, of course, but still, under the right societal conditions, people can be motivated to repent. I also think of Emperor Theodosius, who only repented after forceful rebukes by St. Ambrose. Should we not incentivize such righteous actions in our own societies? Of course, we should!
Much temporal good has come from our Holy Mother Church, the Mater et Magistra ("mother and teacher"). In the Middle Ages, for example, during which the power of the Church was at its highest, literacy rates increased, art flourished, the university system developed, laws were better-codified, and the Bible became more accessible to lay people. This was all due to popes leading as shepherds and flocks following like sheep. The laypeople of the period, I think, had a greater appreciation for hierarchy and understood that, if left to its own devices, society would collapse. Remember that God had to come down to His people; His people did not go up to Him. Society, today, unfortunately, detests anything other than near-absolute egalitarianism.
Now, imagine for a moment, a Papacy unhindered by societal pressures, free to guide its flock to the fullest, and empowered to pave a beautiful kingdom for Christ's return! Shepherds like Pope Alexander VI understood and promoted this possibility. Unfortunately, because of the rise of democracy and secularism, the temporal honors historically afforded to popes have dwindled. Now, an Alexander would be almost universally despised, and for no reason!
The Pope, the Vicar of Christ, the Visible Shepherd, the Father of princes and kings, the pinnacle of civilization -- he needs us. Pray for him and forever support him in every way possible. And pray that the Church may be empowered to pave the way for Her King.
"[W]hat is certain is that the ruins and traces of the Holy Empire are all about us. An understanding of its history and continuing influence is key to understanding the practical implications of the Social Kingship of Christ — which idea, in so many ways, is the ideal successive Emperors and their loyal subjects sought to follow on Earth, and without which, as Pius XI teach[es] in Quas primas, real peace is impossible." -- Charles A. Coulombe