Peter singled out with a special authority I do not dispute. He was an apostle. The apostles did have a different role in the church than we do today, and Peter was reputed to have been a pillar (one of at least 3). But you do not see him as being above the others. He was just a leader and apostle in the early church.
Further evidenced by his own writing: He saw himself as just another elder
1 Peter 5:1
The elders which are among you I exhort, who am also an elder, and a witness of the sufferings of Christ, and also a partaker of the glory that shall be revealed:
5:2 Feed the flock of God which is among you, taking the oversight thereof, not by constraint, but willingly; not for filthy lucre, but of a ready mind;
5:3 Neither as being lords over God's heritage, but being ensamples to the flock. 5:4 And when the chief Shepherd shall appear, ye shall receive a crown of glory that fadeth not away.
"But you do not see him as being above the others. He was just a leader and apostle in the early church."
No, not above the others, but, as I said, among the others. For this reason as well as the fact that "Old Rome" was the original seat of the Empire, the Ecumenical Councils decreed that the Bishop of Rome, as successor to +Peter, is the "first among equals" of the bishops of The Church. Orthodoxy takes that position to this day, but of course for now the Bishop of Rome is in schism from the Church in the East.