That is correct.
Beginning from the baptism of John, unto that same day that he was taken up from us, must one be ordained to be a witness with us of his resurrection." -- Acts 1:21-22 "Wherefore of these men which have companied with us all the time that the Lord Jesus went in and out among us,
After praying, lots were thrown and Matthias had been with Jesus since his Baptism, which was one of the criteria Peter talked of before lots were thrown. If that was one of the criteria, being with Jesus from the baptism until he was chosen, Paul does not fit the criteria of being with Jesus since his baptism, so I still do not have an answer, by Peters own prayer, he stated the criteria for being chosen an apostle.
Peter said you had to be with Jesus since he was baptized. Prayer was said for God to choose by lot. Matthias was chosen and became the 12 apostle. As 12 were necessary because Jesus originally chose 12. Not 13.
Were prayers said and lots thrown to have Paul become a apostle..Or was Paul self anointed...One would not have to be an apostle to be sent to the gentiles.
It seems Paul wanted that title and just took it. He was not with Jesus since his Baptism by John.