Whilst this is true the claims of the papacy to secular rule never really gained the same acceptance as the claims of the Caliphate. Sunni Muslims's seemed to have pretty universally recognized the Caliphs claims (at least in theory) whilst Popes who were too aggressive ended up dead like Pope Boniface VIII. Theologians like John of Paris argued that Papal and kingly power were both independently established by God in On Royal and Papal Power.
I would disagree, the Papacy was able to assert this authority and this was readily accepted until Boniface VIII and even after to a degree such as the case with King Peter Anscarid in Castile, deposed by Papal Interdiction and Papal support of Henry Trastamara. Even so, an extremely anti-Boniface VIII intellectual as Dante, accepted clearly the ability and authority of the Papacy to distribute the Imperial title in his "De Monarchia". The opinion in Europe, perhaps outside of Paris, was that the Papacy held the rights regarding the Translatio Imperii, none would have claimed that the feudal lords of Europe restored Rome individually, rather this was done only through Papal edict. In terms of other examples of the Papal authority as an accepted quantity, can be seen in the submission of John Lackland to Innocent III or the victory of the Papacy over the Hohenstaufen.
We also should note the words of Late Medieval rulers regarding the Western Schism, from John 'the Fearless' future duke of Burgundy: 'Bayezid was courteous, but made clear his intentions; to make his horse eat corn upon the seat of Saint-Peter. He knew that victory was imminent as the lands of Christendom were divided between two rulers! These Saracen had made great marvel at the suffering ...was to be felt by the Lords and Kings for this division of leadership.' In that sense, there was an understanding from both the Turks and the monarchs in Europe as to their legitimate ruler and the lifeblood of their states. Another point is when we look at who acquired letters of correspondence for Catholic alliances such as the famed letters exchanged between the Papacy and the Mongol Empire, were made directly to the Papacy, not to the princes of Christendom.
Regarding Boniface VIII, his position was one of the conservative, defending Papal and Ecclesiastical privileges which had been set centuries prior. To his defense, he possessed the Donations of Constantine and Pepin, the Translatio Imperii, the powers of excommunication and interdiction which absolve the subjects of loyalty to their rulers, the Papal victory over the Byzantine Empire in Italy and many centuries of scholastic evidences and statements regarding the Papal authority;
'.... If anyone teaches you anything contrary to that spoken of by the Catholic Church/Papacy (in context Papacy), that has been bestowed based upon that tradition received by the Apostles, the fathers, the synods, and that which is made tradition tot his day (ex cathedra), let that person be anathema. Even if an Angel, even if the Emperor tells you something other than what you have received through tradition, close your ears to him.' -John Damascene
'Brothers and fellow Bishops, I most diligently commend the Bishop of Amiens, your procurator and advocate, to you kindness. Indeed he worked against aforesaid privilege in the Roman Curia with all of his might, but achieved nothing; so he has tried to avenge his lost pride here. As you see, the effort and expense has worn him out. But I declare before you all, that our mission is not to revoke said privilege, nor to molest the friars whom you decry, but rather to confirm the privilege itself; because this is the only healthy limb we have found. On this account they will not be hindered but rather favored. Therefore we wish that the privilege remain as it was originally decreed.It is a pity that the masters of Paris are not present to see their stupidity shown in the light of day. With impious and rash temerity they presumed to interpret the aforesaid privilege as if the Curia had conceded it without necessary deliberation. They must know the Roman Curia has feet not of feathers but of lead. Indeed these masters think we consider them wise; on the contrary, they are more foolish than all the fools, because they have filled not only their heads but the whole world with their pestilential doctrine. Nevertheless, in virtue of the authority that has been delegated to us to this end, we revoke and annul whatever any one has attempted against the aforesaid privilege. Otherwise, every privilege granted by the Holy See could be similarly nullified by the activities of the monarchs.' -Boniface VIII, he clearly stated that the privileges he would die for, was the original situation or status quo, it was wars that were waged and the need for new taxes that saw the progressive changes take form within the French monarchy and the English monarchy.
As an addendum, Boniface VIII clearly made the point that according to the laws regarding taxation and confiscation of resources, that the legal opinion was that this was allowed by tradition by consent. Boniface VIII thus made the clear offer to the French crown, to give the necessary funds they sought for the wars, but only if the French crown asked and this would be in line with the traditions in this respect. Rather, the French crown attempted to nullify its past legal precedence and make edicts in contrary to its predecessors. So it was not that Boniface VIII was some sort of lone absolutist, but that the new situation was such that the Papacy, defending its traditional power structure, was bested then, when prior it had generally held the upper-hand over the princes.
Sorry to go off topic in this thread....