Baloney.
The subjugation is coldly spelled out as both spiritual & temporal. The spiritual, salvific subjugation is achieved thru monoply on censecrating power - the sacraments.
Read it again, it is truly bodacious:
"We venerate this Church as one, the Lord having said by the mouth of the prophet: 'Deliver, O God, my soul from the sword and my only one from the hand of the dog.' [Ps 21:20] He has prayed for his soul, that is for himself, heart and body; and this body, that is to say, the Church, He has called one because of the unity of the Spouse, of the faith, of the sacraments, and of the charity of the Church. This is the tunic of the Lord, the seamless tunic, which was not rent but which was cast by lot [Jn 19:23-24]. Therefore, of the one and only Church there is one body and one head, not two heads like a monster; that is, Christ and the Vicar of Christ, Peter and the successor of Peter, since the Lord speaking to Peter Himself said: 'Feed my sheep' [Jn 21:17], meaning, my sheep in general, not these, nor those in particular, whence we understand that He entrusted all to him [Peter]. Therefore, if the Greeks or others should say that they are not confided to Peter and to his successors, they must confess not being the sheep of Christ, since Our Lord says in John 'there is one sheepfold and one shepherd.' We are informed by the texts of the gospels that in this Church and in its power are two swords; namely, the spiritual and the temporal. For when the Apostles say: 'Behold, here are two swords' [Lk 22:38] that is to say, in the Church, since the Apostles were speaking, the Lord did not reply that there were too many, but sufficient. Certainly the one who denies that the temporal sword is in the power of Peter has not listened well to the word of the Lord commanding: 'Put up thy sword into thy scabbard' [Mt 26:52]. Both, therefore, are in the power of the Church, that is to say, the spiritual and the material sword, but the former is to be administered _for_ the Church but the latter by the Church; the former in the hands of the priest; the latter by the hands of kings and soldiers, but at the will and sufferance of the priest."