Orthodoxy neither denies the primacy of St. Peter nor the fact that he is often singled out in the Gospels and early Church as being the foundation and key-holder of the Church. However, these prerogatives must be understood correctly by reviewing other patristic writings, which provide further clarification on the meaning of the quotes you provided.
St. Augustine:
“I mean, to show you that it is the Church which has received the keys of the kingdom of heaven, listen to what the Lord says in another place to all his apostles: ‘Receive the Holy Spirit;’ and straightway, ‘Whose sins you forgive, they will be forgiven them; whose sins you retain, they will be retained’ (Jn 20:22-23). This refers to the keys, about which it is said, ‘whatever you loose on earth shall be loosed in heaven, and whatever you bind on earth shall be bound in heaven’ (Mt 16:19)” - St. Augustine, Sermon 26, 1-4.
And
“Its clear, you see, from many places in scripture that Peter can stand for, or represent, the Church; above all from that place where it says, To you will I hand over the keys of the kingdom of heaven. Whatever you bind on earth shall also be bound in heaven; and whatever you loose on earth shall be loosed in heaven. (Mt. 16:19). Did Peter receive these keys, and Paul not receive them? Did Peter receive them, and John and James and the other apostles not receive them? Or are the keys not to be found in the Church, where sins are being forgiven every day? But because Peter symbolically stood for the Church, what was given to him alone was given to the whole Church. So Peter represented the Church; the Church is the body of Christ” – St. Augustine, Sermon 149, 6-7.
These quotes reveal that the entire Church, through her shepherds, receive what was given to St. Peter. This is possible because Christ singled out St. Peter in the apostolic commission in order to represent the oneness of the Church and its gifts and not to give St. Peter something that isn’t given to the rest of the Church. This belief is mirrored in other patristic writings, such as those of St. Cyprian of Carthage and St. Gregory of Nyssa, which see in St. Peter the beginning of the oneness of the episcopacy. This is also implicitly affirmed by the fact that the Fathers never apply Matt. 16:18-19 to Rome in an exclusive sense, but extend its interpretation to the entire Church, built firmly on the rock of St. Peter through confession of the true Faith.
As you know, the ancient Church looked on any apostolic church as being a solid benchmark for determining orthodoxy. The Roman Church, being the capital of the ecume and possessing a double apostolic origin in Sts. Peter and Paul, is often singled out for emphasis as bearing witness to the common Tradition. However, this emphasis cannot be interpreted as claiming for the Roman Church something the other apostolic Churches lack until you explain why in other locations, St. Irenaeus, Tertullian, St. Vincent of Lerins, St. Augustine and other Fathers direct seekers to other apostolic churches with the same confidence that they do the Roman See.
God bless,
Adam