Hi Shirahui !
Concerning the belief that Gregory the Great would hold to an equalization of sees, I’d propose as a source himself.
You offered IMHO a great quotation in post 78. You added there an interpretive key, according to which the pope just would mean that anybody in case of fault is subject to anybody’s correction, if I get.
Let’s read the context:
the Byzacene primate had been accused on some charge, and the most pious Emperor wished him to be judged by us according to canonical ordinance. But then, on the receipt of ten pounds of gold, Theodorus the magister militum opposed this being done. Yet the most pious Emperor admonished us to commission some one, and do whatever was canonical. But, seeing the contrarieties of men, we have been unwilling to decide this case. Now, moreover, this same primate says something about his own intention. And it is exceedingly doubtful whether he says such things to us sincerely, or in fact because he is being attacked by his fellow bishops: for, as to his saying that he is subject to the Apostolic See, if any fault is found in bishops, I know not what bishop is not subject to it. But when no fault requires it to be otherwise, all according to the principle of humility are equal. Gregory the Great Letter IX, 59
The matter is specifically about whether the Pope should formally judge the primate of Byzacene, a North African region. That enlightens what Gregory means when he unequivocally states that all bishops ( patriarchs too are bishops BTW ), in case of fault, are subject to him.
If doubts remain on what Gregory signifies by that we can read the following:
TO bishop of Prima Justiniana ( in present day Serbia )
as regards the present, by the authority of the blessed Peter, Prince of the apostles, we decree that, the decrees of your judgment being first annulled and made of none effect, thou be deprived of holy communion for the space of thirty days, so as to implore pardon of our God for so great transgression with the utmost penitence and tears. But, if we should come to know that you have been remiss in carrying out this our sentence, know thou that not the injustice only, but also the contumacy, of your Fraternity will have to be more severely punished. Book III, letter 6
To the bishop of Larissa ( Greece ),
*though you ought to have been deprived of the communion of the Lord’s body, for that, setting at naught the admonition of my predecessor of holy memory, whereby he exempted him ( *he’s referring to the bishop of Thebae **) and his church from the jurisdiction of your authority, you have again presumed to retain some jurisdiction over them, yet we, decreeing more humanely, and still allowing you the sacrament of communion, decree that your Fraternity shall abstain from all exercise of the jurisdiction formerly held by you over him and his church; but that, according to the written instructions of our predecessor, if any case should possibly arise, whether touching the faith, or criminal, or pecuniary, against the aforesaid Adrian our fellow priest, it be either taken cognizance of, if the question be a slight one, by those who are or may be our representatives in the royal city, or, if it be an arduous one, it be brought hither to the Apostolic See, to the end that it may be heard and decided before ourselves.
But, if you should attempt at any time, on any pretext or by any surreptitious device, to contravene these our ordinances, know that we decree you to be deprived of holy communion, and not to partake of it except at the close of your life, unless upon leave granted by the Roman pontiff. Book III, letter 7
To the bishop of Sabinianus of Jadera ( in present day Croatia ),
*the inhabitants of the city of Epidaurus have most urgently requested us to restore to them Florentius, whom they allege to be their bishop, asserting that he was driven into exile invalidly by the mere will of the bishop Natalis. And so, if your Fraternity has any knowledge of his case, please to inform us accurately by letter. But, if so far you have no knowledge of it, make enquiry, and report to us, that we may be able, with the Lord’s help, to deliberate with full knowledge before us as to what should be determined concerning him.*VIII, 10
(Quotes from
newadvent.org/fathers/ )
It appears that by exercising his authority Gregory himself IMHO shows us what he means stating that all bishops in case of fault are subject to the Apostolic See.