What exactly is meant by "universal ruler" here? You have explained what it does not mean, but what are the positive ideas/parts that provide some content for the title? The Patriarch of Constantinople has some abilities that other bishops do not have (in spite of the idea that he is theoretically equal to all other bishops in a collegial system), such as presiding over councils, settling disputes which cannot be settled at the local level, and even in rare occasions helping to resolve disputes in which the disputants have not directly petitioned him for help. Is this the type of thing you mean by "universal ruler..."
Yup thats what I mean. I would add that the most usual implementation of this role was writing letters to enforce [ie ensure lawful observance of] ecumenical councils, canons, local decisions ectetera. It is a pastoral role
or ministry of oversight
. The popes wrote hundreds of such directional letters to Eastern dignitaries. This was their "universal rule."
Perhaps the Church supplies some extraordinary
"jurisdiction" from custom, as it does for patriarchs and metropolitans, that is used in particular circumstances. (The circumstances are what we have listed.) However, this is not real
jurisdiction, since it is granted from the Church via custom to be used at certain times alone, and its sole purpose is to ensure administrative cohesiveness. Its true that in the original constitution of the Church each bishop is equal in power. (The preeminence of Peter which passed to those who succeeded to his chair in Rome was not jurisdictional power but mere honorific leadership.) But in practice the Church has the right to grant more power to bishops. The original leadership role that the bishop of Rome had was amplified when the Church was made to correspond to the structure of the Empire. Pre-Christian Eternal Rome was called "caput mundi" or head of the world. Likewise this title passed on to the Roman church. (The pope was in fact called universal "head" more than universal "ruler.") These terms were common parlance back then. The original meanings were quite different from how the high medievals interpreted them. It was also understood in ancient times that whoever is the first bishop, whether the metropolitan in his area or the patriarch in his area or the pope in his area, is de facto called "head" and "ruler." It is equivalent to our modern world "president."
In addition to the tasks that we listed above I would also like to add that, in cases of necessity such as when patriarchs were not doing their job or when a patriarch became a heretic and no higher authority could take care of the situation
, the pope could excommunicate people outside his own patriarchate. This should come as no surprise though since even regular patriachs could excommunicate other patriarchs. For instance Alexandria and Constantinople could issue mutual excommunications against each other. Likewise the pope could do the same. Indeed he was responsible for it. Whereas other patriarchs were not responsible in the same sense for other patriarchates. The pope's "universal rule" was to care for all the churches.