Not to wade too far into weeds, but the spiritual and temporal authority of Rome (ie what is now called Church and State) were considered separate entities. When the temporal authority collapsed, the spiritual authority preserved its content and form and then simply waited for conditions to improve.
Τhus the various attempts to re-manifest a Holy Roman Empire in Europe across the subsequent centuries.
Rules? Membership of an organisation is nothing like being a subject of an empire. Its purely voluntary. It has less power over its members than employers do. It cannot tax, only ask for donations. Its not even Rome - its run by people from all over the world. It controls only a tiny bit of territory.
question - do the representatives and elected members of a democracy rule?
Rule, under the definition of control or dominion, is "The exercise of authority over a region or people, or the duration of a monarch's reign (e.g., colonial rule)", the word rule is often applied to mean what the leaders of a democracy do, but in most cases rule seems too strict if taking the normative definition, thus there is an informal usage that applies to democratic leadership.
The President is often referred to as a ruler, corporate elites are often referred to as rulers; in these informal groups of rulers I think it is reasonable to place the Catholic Church.
I am agreeable to say they do not rule, but then we must admit they exert a great influence, in many places and ways greater than the governments that officially hold power.
Τhus the various attempts to re-manifest a Holy Roman Empire in Europe across the subsequent centuries.