One compelling biblical fact that points clearly to Simon Peter’s primacy among the 12 Apostles and his importance and centrality to the drama of Christ’s earthly ministry, is that he is mentioned by name (e.g. Simon, Peter, Cephas, Kephas, etc.) 195 times in the course of the New Testament. The next most often-mentioned Apostle is St. John, who is mentioned a mere 29 times. After John, in descending order, the frequency of the other Apostles being mentioned by name trails off rapidly.
When the names of all the Apostles are listed, Peter is always first. Judas Iscariot, the Lord’s traitor, is always listed last (cf. Matt. 10:2-5; Mark 3:16-19; Luke 6:14-17; and Acts 1:13). Sometimes Scripture speaks simply of “Simon Peter and the rest of the Apostles” or “Peter and his companions” (cf. Luke 9:32; Mark 16:7; Acts 2:37), showing that he had a special role that represented the entire apostolic college. Often, Scripture shows Simon Peter as spokesman for the entire apostolic college, as if he were the voice of the Church (cf. Mat. 18:21; Mark 8:29; Luke 8:45; Luke 12:41; John 6:68-69).
The Primacy of Peter - from Where's That In The Bible, by Patrick Madrid.
That Peter was first among the Apostles and the primary spokesman of the Apostles is without a doubt --- but only until Paul came on the scene and the Gospel began to go to the Gentiles. Peter was the Apostle to the Jews and first leader of the Jewish Church in Jerusalem. His ministry was to the Jews, not the Gentiles.
When Paul came on the scene and the Gospel began to be presented to the Gentiles, he overshadowed Peter and he became the primary Apostle. Aren't the Pauline Epistles, particularly Paul's Letter to the Romans, the heart of the Gospel to the Gentiles? Wasn't the Roman Church a Gentile Church? That Paul spent years in Rome and died there is a fact of history and attested to in the New Testament. Where Peter died is a matter of speculation and legend without New Testament attestation.
Paul is, by far, the most prolific writer in the New Testament. Does that make him the most important?
There is no doubt Peter is pre-eminent among the Apostles but he was not their undisputed leader. The Apostles acted collegiately and at times directed Peter on his mission. They could not have done that if he was the one supreme leader.
Acts 1:
23 And they put forward two, Joseph called Barsab'bas, who was surnamed Justus, and Matthi'as.
24 And they prayed and said, "Lord, who knowest the hearts of all men, show which one of these two thou hast chosen
25 to take the place in this ministry and apostleship from which Judas turned aside, to go to his own place."
26 And they cast lots for them, and the lot fell on Matthi'as; and he was enrolled with the eleven apostles.
Now, of course, the Holy Spirit was working within the Apostles but it was they who cast lots and selected Matthias. The Pope chooses his Cardinals. The Apostles chose their replacement. They, not Peter. Peter was simply one of them, an equal one.
"... Often, Scripture shows Simon Peter as spokesman for the entire apostolic college, as if he were the voice of the Church (cf. Mat. 18:21; Mark 8:29; Luke 8:45; Luke 12:41; John 6:68-69)."
Often, but not always. That is a critical difference.
Acts 8:14
Now when the apostles at Jerusalem heard that Sama'ria had received the word of God, they sent to them Peter and John,
Peter was subordinate to the group.
Acts 15:19 Therefore my judgment is that we should not trouble those of the Gentiles who turn to God,
James made the binding judgment because it was in his territory. In this instance Peter was acting as one of the Apostles, was a member of the audience, and was definitely not in charge.
Luke 16:15 15 And he said unto them, Ye are they which justify yourselves before men; but God knoweth your hearts: for that which is highly esteemed among men is abomination in the sight of God.
Luke 14:11 11 For whosoever exalteth himself shall be abased; and he that humbleth himself shall be exalted.
Paul many times stated that he was not worthy to be apostle where others fought to be seated next to Christ in the Kingdom.