73 Don,

I agree with all of your 1-4 points except your last #4 point which I assume should have been #5. I believe that Paul claims to be one of the 12 apostles.

Rom 1:1 Paul, a bond-servant of Christ Jesus, called as an apostle, set apart for the gospel of God,
Rom 1:2 which He promised beforehand through His prophets in the holy Scriptures,
Rom 1:3 concerning His Son, who was born of a descendant of David according to the flesh,
Rom 1:4 who was declared the Son of God with power by the resurrection from the dead, according to the Spirit of holiness, Jesus Christ our Lord,
Rom 1:5 through whom we have received grace and apostleship to bring about the obedience of faith among all the Gentiles for His name’s sake,

In Romans 1:1-5 Paul declares that he is an apostle of Jesus Christ. Paul then goes on to declare the gospel that the foundational apostles were specifically chosen to be witnesses of: the resurrection of Jesus Christ. Then Paul states that it is through Jesus and his resurrection that “we” have received grace and apostleship. In verse 5 Paul aligns himself with the other apostles that Jesus specifically hand picked to be witnesses of the resurrection.
While there were others in the church who were apostles and there were others who were leaders (James for example), there were only 12 who were picked by Jesus to be eyes witnesses to the resurrection and these were all to be apostles of Jesus Christ. No other apostle claimed to be picked by Jesus nor did they claim to be an apostle of the resurrection. Paul claimed these things.
Paul also claimed that he was equal to the other apostles picked by Jesus:

2 Co 11:5 For I consider myself not in the least inferior to the most eminent apostles.

Paul also claimed that he had the signs of a true apostle:

2Co 12:11 I have become foolish; you yourselves compelled me. Actually I should have been commended by you, for in no respect was I inferior to the most eminent apostles, even though I am a nobody.
2Co 12:12 The signs of a true apostle were performed among you with all perseverance, by signs and wonders and miracles.

No other apostle in the NT claimed to be picked by Jesus. No other apostle claimed to be an apostle of the resurrection. No other apostle claimed to be equal to the other apostles of Jesus Christ. No other apostle claimed to have the miracles to prove that he was a true apostle. No other apostle even had to have miracles to prove that he was a sent one from the church. But an apostle of Jesus Christ had to be picked by Jesus, had to be a witness to the resurrection and had to have the “signs” of that apostleship.

While this is not a hill to die on nor is it a necessary thing to believe, I myself accept Paul’s claim to be an apostle of Jesus Christ one called to be a witness of the resurrection and with the evidence of his apostleship supplied by his signs, wonders and miracles and his great revelations from Jesus Christ. Because of this, I believe that Paul was personally picked by Jesus as the 12th apostle of the resurrection and Matthias, although he was an apostle of the church, was not one of the twelve picked by Jesus.