Perhaps you should read what else Blessed Cyprian had to say before you comment.
’...thou art Peter and upon this rock I will build my Church’ IT IS ON HIM THAT HE BUILDS THE CHURCH , and to him that he entrusts the sheep to feed. And although he assigns a like power to all the apostles, yet he founded a single Chair, thus establishing by his own authority the source and hallmark of the (Church’s) oneness...If a man does not fast to this oneness of Peter, does he still imagine that he still holds the faith. If he deserts the Chair of Peter upon whom the Church was built, has he still confidence that he is in the Church? Cyprian, De Unitate Ecclesiae (Primacy text), 4 (A.D. 251).
There are more of these writings also.
I wish you a Blessed Day!
Virtually all the writings in the New Testament are addressed to specific churches,
1Co 1:2 To the church of God which is at Corinth,
1Co 16:19 The churches of Asia greet you.
Col 4:15 Greet the brethren who are in Laodicea and also Nympha and the church that is in her house.
Eph 5:23 For the husband is the head of the wife, as Christ also is the head of the church,
The church is subject to Christ - Eph 5:24 But as the church is subject to Christ,
Christ will present the church to God - Eph 5:27 that He might present to Himself the church in all her glory,
One universal Church with a Pope handed down is not only historically inaccurate but it is scripturally inaccurate. The Orthodox don't even agree with this. Scripture at times refers to the church as all encompassing. At other times it is used independently. I think that is how the early church fathers saw it as well.