Jesus gives Peter the Keys to kingdom. Peter is head of Catholic Church.
More than that, much of the Orthodox's positions are a manifestation of the fact that they are controlled and have been controlled by the State, namely the Greek Emperor or the Tsar, and his interest. He opposed the power of Saint Peter and found policies and clergy within the Greek church that were amenable to his political goals, at the expense of true orthodoxy. I don't see how anybody can read the history of the Eastern church and not come away with this impression.
They have no systematic metaphysics and their arguments tend to be strongly anti-intellectual in their final analysis. Their entire schism is self-justifying, whereas we actually have valid and satisfactory arguments for the Petrine primacy and the rest of our positions and beliefs at variance with the East.
Further, many of their current beliefs are contrary to their former beliefs, which were closer to ours. They formerly believed in the All-Holiness of the Theotokos, or rather the Immaculate Conception, but they changed their tune after Pius IX defined the dogma in 1854. They also believed in the Tollhouses, but they now claim that Purgatory is an invention of the West.
Their position relies, at root, on the idea that Greek theologians' words are more valuable and authentic than Latin theologians, period. Ultimately, they therefore put the Christian State at the head of their church, since they could never otherwise compete with the claim of the Bishop of Rome to be the successor of St Peter with supreme and immediate jurisdiction in the whole Church. They are and have been morally and politically compromised.