A couple of points here…
First, the king of our kingdom is not the pope, but rather Christ. We even have a feast day called Christ the King to remind us of this fact every year. So as far as the monarchy thing goes, you are correct in that it IS a monarchy, but I’d argue that you are incorrect as to WHO that is.
Second, the office of Peter is that of a prime minister. We see in the OT that there was a prime ministry separate from the kingship (Is 22:22). So, a one-to-one comparison is not really appropriate.
Third, since Christ appointed Peter to “tend to His sheep”, gave him “the keys to the kingdom to bind and loose on earth”, and commanded him to “strengthen his brethren [apostles]”, then I’d say the pope has a DUTY to fulfill the ministry given him through the command of Christ.
Fourth, You assume too much in giving the Pope such lordship. The pope cannot bind the Catholic believer to any whim of his 9unlike a mlitary ruler and king can). There is no restriction, no denial of the average man’s free will here. If anything, the pope is the most restricted individual in the whole church. He is bound by the Tradition of the Church.
Fifth, saying that the pope is infallible is not saying that he is not sinful or cannot err in his own spiritual walk. It is a guarantee that he will be protected from teaching error on matters of faith or morals when teaching in an official capacity as visible head of hte church (ex cathedra). So, in saying that King Saul was sinful and prideful, therefore we should not trust a Pope, I’d say “Paul, Luke, Matthew, Mark, and John were all sinful men, yet YOU trust your eternal future to their writings.” You may say, “Well, I believe that the Holy Spirit used these sinful men to teach infallible truths.” I’d then say, “Precisely my point.”