Like other times in Peter's life, he stepped out in front of God in saying "Let's see which of these two God has chosen." In reality God had chosen neither Joseph or Matthias, but Saul. In 1 Cor. 9:1 and throughout 2 Cor. Paul identifies himself with the 12 by saying "Am I not an apostle; have I not seen Jesus?" In Galatians, he claims to have been taught by Jesus just as were the other apostles.
We must understand that inspiration of Scriptures does not mean every thing said in the Scriptures is true, but that everything said in the Scriptures was actually said. It was not true, for example, that Jesus would never be killed on the cross. But it is true that Peter said it in Matthew 16:23, in response to which Jesus called Peter Satan.
Peter also got out before God on the Mt. of Transfiguration offering plans to build three altars. God had to tell Peter to shut up and listen to Jesus. Even Paul had to rebuke Peter in Galatians 2:11-14.
Peter was a great man and a pillar of the church, but it is not he who is inerrant; it is the Scriptures themselves.
After the day that Peter led the other 10 apostles to cast lots for Judas' replacement, no one ever hears from Matthias again. We know nothing of him. Paul, on the other hand, who was clearly God's choice as an apostle became the central figure in the propagation of the gospel especially to the Gentiles. He wrote more than half the book in the New Testament, He took the gospel on at least three missionary trips throughout their world, he was stoned, drowned, beaten, imprisoned, starved, hated, loved. He raised (through God's power) a young man who fell to his death in a church meeting. In short, Paul made full proof of his apostleship.
Matthias must have been an honorable man, but he was not (in my opinion) God's choice for Judas' replacement. I believe when we get to heaven and walk around the walls of the New Jerusalem, we will see the names of the 12 Apostles of the Lamb. One of those stones will bear the name of Paul, Apostle to the Gentiles.