The short answer is yes. The Patriarchs have the authority to write new canons, change the liturgy, and pass judgement on cases. They can call councils, depose Bishops and priests under them, judge and regulate monasteries, and discipline clerics, monks, and nuns, as well as the lay faithful. All these things they have done in the past.
If so, why is there a rejection of the concept of Papal supremacy?
Because the Pope claims the jurisdiction of final appeal over all these things for the whole Church, and the right to intervene in the affiars of every Church, and not just the lands under him as Patriarch in Rome.
they aren’t able to declare an aspect of doctrine infailably if that’s what you mean...
When the Patriarch of Constantinople presided over the Palamite Councils of 1341-1351, did he think they were not definitively settling doctrine under his guidance, but just issuing opinions?
When Dositheos, Patriarch of Jerusalem, made his profession of faith against Cyril Lukaris and Co. in 1672, did he think he was not definitively setting forth the position of the Orthodox Church against the inroads of Calvinist heresy, but just offering an opinion which one could take or leave at will?
One of the areas that could use much more examination by both sides here is the powers inherent in the office of Bishop, and the level of guidance provided by the Holy Spirit in preventing Bishops from falling into teaching heresy, and especially to the Chief Bishops in the various parts of the world - the Patriarchs. I think the position of the Pope vis-a-vis his own powers is quite clear.
However, historically speaking, it also seems clear that it is the will of Christ that the organization of the Church be along heirarchical lines with jurisdictional power resident in the Bishops of the chief cities of the world, and that these Bishops are to oversee the provincial Bishops in lesser towns. It is not an accident that St. Peter went to Antioch and then Rome, and sent St. Mark to Alexandria, while St. John went to Ephesus, St. Jude to Edessa, etc. The purposefully went to the largest cities and set up the Church as operating out of those places and organizing the province from the principal places.
An historical exploration of the actual life and practice of the Church as far as its operations, organization, and planning would do much to illuminate what the Church believes concerning primacy and jurisdiction within the Church.
What also should be explored is the organization of the Church outside the Empire, which would do much to illuminate the intentions of the organization within the Empire which is the nucleus of the Catholic and Orthodox Churches. Not only because it is illuminating, but also because the same issue is faced with relations of both Churches to the Oriental Orthodox and the Church of the East. Especially helpful here is the historical relationship of the Patriarch of Alexandria to the Ethiopian Church (a clearly paternalisitic/Pope-like role being played by the Patriarch towards the Ethiopians), the Patriarch of Antioch to the Persian Church in Seleucia-Ctesiphon (and the raising of that Church to Patriarchal dignity in the Nicean-Ephesian era), and the Patriarch of Constantinople to the Russian Church.
While the Church has always honored its great thinkers as Doctors, even while they were alive (thus the reverence accorded to St. Basil and St. Augustine, for example, neither of whom ever held a major see), those same Doctors lived under a system where they were under the Bishop of the chief city. St. Augustine may have been the light of Africa, but the Archbishop of Carthage was its ruler, and clearly held jurisdictional power over all the other Bishops including Augustine, while the Archbishop of Carthage in his times, Caecilian, himself answered to Rome.
When the de facto life of the Church is brought out de jure, the beliefs of the Church will be much more clear to everyone, making the self-understanding of the Pope and the various Patriarchs more understandable and hopefully integrable as a whole, so that all the Christian Churches might commune in peace with one another as one flock.
As an example of the types of subjects which truly need explored by each of the Church in gaining self-understanding to be able to explain in words as their beliefs based on the actions they actually undertake, there is the following comment from Vatican II's "Lumen Gentium":
N.B. Without hierarchical communion the ontologico-sacramental function [munus], which is to be distinguished from the juridico-canonical aspect, cannot be exercised. However, the Commission has decided that it should not enter into question of liceity and validity. These questions are left to theologians to discuss-specifically the question of the power exercised de facto among the separated Eastern Churches, about which there are various explanations.
To provide a biref synopsis, this paragraph is stating that the Roman Church, while not being in Communion with the Eastern Churches, nevertheless accepts that a true power of jurisdiction resides in the Bishops of those Churches which the Latin Bishops and the Pope are bound to recognize and follow the rulings of when cases come from those Churches to them (such as canonical rulings, acts of dispensation from laws, creation of new laws, etc.). For example, a couple who need a dispensation from the laws of the Church to be married obtain one from their local Orthodox Bishop. Later, they come to live in the west and take up Catholicism. The Catholic Church recognizes in practice the dispensation granted by the Orthodox Bishop, even though a strict interpretation of Roman Canon Law would say that because the Orthodox Bishop is not in communion with Rome, he does not hold power to dispense from law and therefore the couple is not really married (which would certainly be the ruling for anyone attempting to obtain a dispensation from an SSPX or Thuc Bishop). Therefore the Roman Law does not match the actual Roman practice. Similarly, Roman Law holds that all priests of the Church are forbidden to have marital intercourse after ordination. However, from Rome's view, the East changed this law legitimately for itself in the Quinisext Council in 692, thereby granting a perpetual indult in favor of priests married before ordination. Rome has always felt constrained to recognize this practice when Easterners have returned to union with Rome, even though it is contrary to Rome's self-understanding of the law of priestly continence, and even though at times the Bishops of the Uniate Churches have wanted to readopt the Roman discipline for their priests.
So even the Catholics need to do some self examination, and come to a fuller understanding of what they believe concerning the office of Bishop to intelligently participate in this discussion, because their actual practices do not correspond to their human laws, and the human laws are the actual face the Easterners see when they look at the Vatican jurisdictional apparatus.
Its difficult for East and West to make a union of the Churches before they truly understand all the implications of how they actually live the life of faith. The East accepting the pronouncements of Vatican I as a basis of union, for example, would only be useful if the West accepts and understands what the East thinks those canons are truly saying in how the Church should be organized, as the Uniate Bishops themselves noted when signing the decrees of Vatican I.
In other words, what are the rights and privileges of the Eastern Patriachs (and all Bishops for that matter), and how do those relate to and limit the jurisdiction of the Pope?
“The Patriarchs have the authority to write new canons, change the liturgy, and pass judgement on cases. They can call councils, depose Bishops and priests under them, judge and regulate monasteries, and discipline clerics, monks, and nuns, as well as the lay faithful. All these things they have done in the past.”
Not at all. These powers you refer to are canonically exercisable only by a synod. There may have been Patriarchs who did otherwise, but those acts were uncanonical.
The rest of your post is really very good.
They have done it, but they are not necessarily canonical. The Serbian Orthodox Church just concluded its two-week synod where the major topic of discussion were liturgical changes implemented by one or a small number of bishops. The 'changes' involved keeping the Royal Doors open throughout the service and reading the opening prayers aloud , whereas a priest is supposed to recite them to himself according to the Typikon.
Some priests actually refused to 'radically' change the Orthodox Divine Liturgy and were suspended by the bishop(s). Their graces were also facing open opposition in the monasteries and laity.
The synod agreed to study the changes and see if there is any justification or need for them, but in the meantime, the Synod decreed (with the Patriarch, who presided, casting his vote as all the rest) that the liturgical service will continue as prescribed by the Typikon.
So, while a bishop is the final authority in any church, he is not at liberty to implement changes that are not canonical, just as a police officer is not allowed to break the law (i.e. speeding) unless in the legitimate performance of his duty. Such a police officer makes himself an outlaw; and a bishop who imlplements uncanonical changes ex-communicates himself from the Church.
I think that is an excellent summary of what's at stake.