When I come to Egypt, I tell you, Egyptians really make you feel so good when you land. My uncles and cousins all would tell me things like, "No wonder all of Egypt lighted up." Or "Really, you bring joy to all the Egyptians." And when I see them honoring their mothers, "Oh Lady, you are the lady of all the beloved" or when they talk to their fathers, "There is no one in Egypt who would ever do anything without your permission, baba."
I think you'll find a lot of that exaggerated honorific expressions in the early Church fathers on popular bishops, and Roman bishops one of the most popular.
All fluff and no substance...what can you trust from those eastern Churches after all?
If you personally love and honor someone, wouldn't you give them that special honor to make them feel special? I don't think this was just an Eatern thing. Otherwise I would question the dogmatic characters of Romantic phrases. Are we really that square to require dogmatically the honor of one bishop above all?
Let's consider the fact that way before Nicea, the see of Alexandria was given the duty of sending annual festal letters for the determination of the date of Easter with a spiritual message. Can we conclude from this practice that the honor given to Alexandria was an infallible source of dating Easter, to be dogmatically held by all Christians?
Consider also the fact of the honor St. Athanasius received from St Gregory the theologian in his 21st Oration:
Thus brought up and trained, as even now those should be who are to preside over the people, and take the direction of the mighty body of Christ,  according to the will and foreknowledge of God, which lays long before the foundations of great deeds, he was invested with this important ministry, and made one of those who draw near to the God Who draws near to us, and deemed worthy of the holy office and rank, and, after passing through the entire series of orders, he was (to make my story short) entrusted with the chief rule over the people, in other words, the charge of the whole world
One might think from this letter, it was Alexandria that was in charge of the Catholic Church. But St Basil in a letter to St Athanasius talks about the importance of the see of Antioch during its problems:
No one knows better than you that like all wise physicians, you ought to begin your treatment in the most vital parts, and what part is more vital to all the churches than Antioch? Only let Antioch be restored to harmony, and nothing will stand I the way of her supplying, as a healthy head, soundness to all the body.
So first we have St Gregory saying that St Athanasius was in charge of the whole church, and then we have St Basil easing to St Athanasius to help restore the head of the Church, Antioch. This type of language is not strange. Nor is this "fluff" carry all falsehood, but rather they do carry truth, but not to the point of dogmatizing it for all generations. And to entertain another letter from St Basil, while Antioch is the head, he reiterates that St Athanasius is the Physician of the whole Church:
When I turn and gaze upon the world, and perceive the difficulties by which every effort after good is obstructed, like those in a man walking in fetters, I am brought to despair of myself. But when I gaze in the direction of your reverence; I remember the Lord has appointed you to be thepjysician of the diseases I. The Churches; and I recover my spirit, and rise from the depression of despair to the hope of better things.
And I understand similar honors were later given to Constantinople. So really, I'm not entirely convinced of the praises to Rome to be some sort of dogmatic stick that puts him above l other bishops. His episcopacy is an honor of love, not of Christian duty or dogmatic coercion. The ecclesiology of the West in my opinion is not something that should be imposed on the East, but I find it at a level of theologomenon, unnecessary for the central faith of a practicing Orthodox and Catholic Christian.
So assuming this is a legitimate quote, considering the Monothelite controversy and the suffering St Maximus went through, he found a supporter and a friend in Rome, and the same honor the Cappadocian fathers gave to Athanasius, maximus gave to Rome.