Okay, and what did Irenaeus of Lyons (+ AD 202) say about the Roman see?
But since it would be too long to enumerate in such a volume as this the succession of all the churches, we shall confound all those who, in whatever manner, whether through self-satisfaction or vainglory, or through blindness and wicked opinion, assemble other than where it is proper, by pointing out here the successions of the bishops qf the greatest and most ancient church known to all, founded and organized at Rome by the two most glorious apostles, Peter and Paul, that church which has the tradition and the faith which comes down to us after having been announced to men by the apostles. With that church, because of its superior origin, all the churches must agree, that is, all the faithful in the whole world, and it is in her that the faithful everywhere have maintained the apostolic tradition -- "Against Heresies"
There are many, many other examples from well before Damasus' time.
Ummm...according to Acts 2 the Holy Spirit already had the church running on all cylinders in Jerusalem, not Rome, before Jesus appeared to Paul on his trip to Damascus. Then later in Acts 15 we had the first church council in ... was it in Rome? No it was in ...wait for it ... Jerusalem.
And if you read the details in Acts 15 what they debated on (inclusion of Gentiles in the church) and who spoke up, it looks to me like the one who had the final say that convinced the council was James. Not Peter. Not Paul. Suggesting to me that the other apostles didn't consider Peter to be some Pope Numero Uno like Catholics claim. Nor, for that matter, did they just agree to it because Paul started off saying so (either of which would have backed up Irenaeus' claim that the church was started by Peter and Paul).
But don't let the church history as recorded in the New Testament get in the way of what you believe about church history. When the Bible and tradition contradict, tradition is superior to Council of Trent believers.