Since this is the only writing you post of an actual Apostolic Father; I’ll cite the opening to Ignatius’ letter to the Romans:
Ignatius, who is also called Theophorus, to the Church which has obtained mercy, through the majesty of the Most High Father, and Jesus Christ, His only-begotten Son; the Church which is beloved and enlightened by the will of Him that willeth all things which are according to the love of Jesus Christ our God, which also presides in the place of the region of the Romans, worthy of God, worthy of honour, worthy of the highest happiness, worthy of praise, worthy of obtaining her every desire, worthy of being deemed holy, Or, “most holy.” and which presides over love, is named from Christ, and from the Father, which I also salute in the name of Jesus Christ, the Son of the Father: to those who are united, both according to the flesh and spirit, to every one of His commandments; who are filled inseparably with the grace of God, and are purified from every strange taint, * abundance of happiness unblameably, in Jesus Christ our God.*
This is merely an exhortation & only indicates Roman primacy in the Roman mind? Here’s another exhortation by Ignatius to the church at Philadelphia:
*Ignatius, who is also called Theophorus, to the Church of God the Father, and our Lord Jesus Christ, which is at Philadelphia, in Asia, which has obtained mercy, and is established in the harmony of God, and rejoiceth unceasingly881881 Or, “inseparably.” in the passion of our Lord, and is filled with all mercy through his resurrection; which I salute in the blood of Jesus Christ, who is our eternal and enduring joy, especially if [men] are in unity with the bishop, the presbyters, and the deacons, who have been appointed according to the mind of Jesus Christ, whom He has established in security, after His own will, and by His Holy Spirit.
Ignatius, who is also called Theophorus, to the Church of God the Father, and of the Lord Jesus Christ, which is at Philadelphia, which has obtained mercy through love, and is established in the harmony of God, and rejoiceth unceasingly,882882 Or, “inseparably.” in the passion of our Lord Jesus, and is filled with all mercy through His resurrection; which I salute in the blood of Jesus Christ, who is our eternal and enduring joy, especially to those who are in unity with the bishop, and the presbyters, and the deacons, who have been appointed by the will of God the Father, through the Lord Jesus Christ, who, according to His own will, has firmly established His Church upon a rock, by a spiritual building, not made with hands, against which the winds and the floods have beaten, yet have not been able to overthrow it:883883 Comp. Matt. vii. 25. yea, and may spiritual wickedness never be able to do so, but be thoroughly weakened by the power of Jesus Christ our Lord.*
I like how Ignatius talks describes the rock that the church is established on – a “spiritual foundation” he calls it, not made with hands. Quite interesting?
Ignatius also wrote:
See that you all follow the bishop, even as Jesus Christ does the Father, and the presbytery as you would the apostles; and reverence the deacons, as being the institution of God. Let no man do anything connected with the Church without the bishop. Let that be deemed a proper Eucharist, which is [administered] either by the bishop, or by one to whom he has entrusted it. Wherever the bishop shall appear, there let the multitude [of the people] also be; even as, wherever Jesus Christ is, there is the Catholic Church. It is not lawful without the bishop either to baptize or to celebrate a love-feast; but whatsoever he shall approve of, that is also pleasing to God, so that everything that is done may be secure and valid.
and his Strong Believe in the Real Presence of Jesus Christ in the Eucharist:
Chapter 7. Let us stand aloof from such heretics
They abstain from the Eucharist and from prayer, because they confess not the Eucharist to be the flesh of our Saviour Jesus Christ, which suffered for our sins, and which the Father, of His goodness, raised up again. Those, therefore, who speak against this gift of God, incur death in the midst of their disputes. But it were better for them to treat it with respect, that they also might rise again. It is fitting, therefore, that you should keep aloof from such persons, and not to speak of them either in private or in public, but to give heed to the prophets, and above all, to the Gospel, in which the passion [of Christ] has been revealed to us, and the resurrection has been fully proved. But avoid all divisions, as the beginning of evils.
He sounds very Catholic to me. He did extort other Christians from other Churches because he is after all a bishop. It is no different than Cardinal from another country coming to the US and condemning certain immoralies.