How are Protestants able to discard the belief in the Petrine Doctrine and a 1500-year history so quickly? Do they not believe that Christ left behind a Church with a teaching authority, or what we Catholics call the Magisterium?
The notion of Petrine supremacy seems to overlook the biblical account that James, not Peter, was the head of the earliest church in Jerusalem (see Acts 15). Peter, James, and John were all considered co-leaders, according to Paul (see Galatians 1-2).
While Protestants certainly believe that God left behind teaching authorities such as the apostles and prophets, the final authority rests with the canon of scripture and not with church tradition about Peter or anyone else, especially if that tradition is not well grounded in scripture. Protestants usually understand Matthew 16:18-20 (Jesus’ statement “On this rock I will build my church”) to refer to Peter’s faith as the basis of the church, not necessarily some Petrine personal authority.

But even if it is the latter, the word “rock” here should be translated “shelf of rocks,” and refers to Peter as a person of faith, and those like him having such authority. Peter was certainly not the first great church leader in Rome, historically speaking.
—Ben Witherington III