Well, lets look at the best descriptive passage we have of the broad governance of the Church in the 1st Century. That passage comes in Acts 15. In this passage we see that the Church as a whole gathered to discuss what the Church's official position would be in relation to the Judaizer controversy.
Let's grant to the Catholics for a second that this is essentially the same format that was used at the Council of Nicaea and the following Eccumenical Councils, that those present were primarily Bishops over loosely autonomous regions.
I think we must ask, based on this council, who is the leader of the Early Church? Rome would have you believe that it is Peter. Here are the primary reasons they give.
In vs 7, Peter stands and addresses the whole gathering after there is much debate. He gives testimony regarding the fact that he himself had preached to the Gentiles, and that they believed and their inclusion in the Church was confirmed by the fact that they received the Holy Spirit. The Catholics view this as a sort of declarative stance where Peter speaks in a nacent "Ex Cathedra" fashion.
In vs 12 we see that after Peter speaks "all the assembly fell silent." They take this to be the crowd being silenced by the authority of Peter as the leader of the Church.
In vs 14 they say that James cites Peter's words as being on the same level of authority as the Prophets.
Now... this isn't a terrible argument if those three verses was all that is included in the passage. However, let me poke some holes.
In verse 7, Peter stands up and addresses the council... however he does so by giving testimony, not by proclaiming a judgement. he asks questions, he recounts the story. He doesn't declare or decide ANYTHING in this context.
in vs 12, the assembly does fall silent... but they do not fall silent in response to Peter's authority... they fall silent in order to listen to Paul and Barnabas. The first "And" in vs 12 is the Greek δὲ (de), which is a disjunctive conjunction. This is used to show a contract between what happens before the de and after. However, the second "And" is the Greek καὶ (kai) which is a coordinating conjunction. This shows that the "Falling Silent" is separated as a disjuncted idea from Peter's speech, and joined to "Listening to Barnabas and Paul" as a coordinated idea. They didn't fall silent in response to Peter, they fell silent to listen to Paul.
Paul then gives basically the same testimony as Peter.
Next, it is clear that the Apostles consider Peter's words to be on par with the Prophets. However, Peter also considered Paul's words to be on par with the Scriptures... Paul cites Luke's gospel as Scripture... this is not a unique phenomena with Peter. Furthermore... The text doesn't even say "Peter" or "Simon" it says "Simeon." No where except here and 2 Peter 1:1 (a text that most critical scholars question in terms of being legitimately Petrine in origin... and that has alternate manuscript evidence that actually says Simon instead) does the New Testament anywhere refer to Peter as Simeon. In fact, if we look at Accts 13, there was an associate at Antioch named Simeon who was considered a Prophet who was also a close associate of Paul and Barnabas while he was there? Is it possible that it is actually THIS Simeon rather than Peter who was speaking? Finally it seems to me that rather than simply rest in Peter's authority to declare this doctrinal truth... James actually adjudicates and judges Peter's statement in light of Scripture, which agree with Peter. That is to say that Peter has not only the correctly reasoned approach, but he has Scriptural support. If ANYONE doesn't need Scriptural Support, it's Peter... but even here the Bible is what determines truth vs opinion.
Finally, and this is the deathstroke for Petrine supremacy... James delivers his judgement. Not Peter... James. Why would James deliver his judgement over matters to which Peter testified (alongside Paul, Barnabas, and others) if Peter was the leader of the Church. Should it not be Peter presiding over the council and providing his judgement?
No my friends... it clearly appears from Acts 15 that James is the leader of the fledgling Church, and that in this dispute Peter (if it even is Peter), Paul, Barnabas, and others simply provide testimony that is ultimately judged by the only normative authority... Scripture.