I was speaking specifically of Patriarchates which deal with the primacy of the diocese itself.
The Ukranian and Ruthenian Catholic Churches do not have Patriarchs, they have Major Archbishops. These are slightly lower in rank with regards to primacy than Patriarchs.
The Melkite Catholic Church did already have a bishop. He was actually elected by the local clergy to be Patriarch but was never installed, as the Council of Chalcedon states, that the Patriarch of Constantinople must approve the election before they are installed. Only a Papal veto could override this decision, a veto which never came at the time. After the installation of the new Patriarch of Antioch, the bishop who was elected but not installed gathered his flock and petitioned Rome for reunification. A Melkite Patriarchate of Antioch was then established via the Patriarchate of Rome as distinct from the original Patriarchate of Antioch which was established by Peter, himself.
The Italo-Greco-Albanian Catholic Church also does not possess a Patriarchate.
The Patriarch of the Armenian Catholic Church did have some union with Rome during the time of the Kingdom of Cilicia (which is now Lebanon) during the crusades, but this fell away. During that time, the Patriarchate moved back to it’s historical location of Vagharshapat in Armenia. The Patriarch emeritus retired in Cilicia and was granted the dispensation to also use the title Catholicos, despite not being the Patriarch any more. At the time of the schism and reuinification with Rome, Cilicia still was not a Patriarchate, thus the diocese of Cilicia was promoted to the Armenian Catholic Patriarchate of Cilicia by Pope Benedict XIV in 1742.
The Maronites were actually the first to petition the Holy See for a Patriarchate. This even predates the Great Schism. In the 7th century, the Maronite community was being neglected by the Patriarchs of Antioch. Because of this, they elected their own Patriarch. The Eastern Church saw it as a schism and an usurpation of the hierarchy of the Church. The Maronites petitioned the Pope for support and Pope St. Sergius I mediated the conflict by creating the Maronite Patriarchate of Antioch separate from the Patriarchate of Antioch. In this case, the Patriarchal succession stemmed from the Patriarch of Rome in the elevation of the Maronite’s principle diocese to a Patriarchate in a similar way to the Melkite and Armenian Catholic Churches.
The only Eastern Catholic Rite which was headed by a Patriarch at the time of attempted reunification was the Syraic Catholic Church. This is why the current Patriarch is given Pimacy over the other Catholic Patriarchs. Unfortunately, the declaration of unification was not official for 120 years and during that time, the Orthodox Church had excommunicated the Patriarch and his successors and installed a new Patriarch. Thus the separate Syraic Catholic Patriarchate of Antioch was recognized by Pope Pius VI in 1783.
God Bless,
Br. Ben, CRM