You only need read the OP.
Does that count as a submission point?
Yes, that has been pointed out to me now thank you. If memory serves me right, the verse was not in the OP originally so I was unaware that you updated it.
Now that I have the verse, I can go back to the originally question and purpose of this thread and give my answer (again).
No, the Church did not go wrong. The reason being all you have to do is read the rest of the verse. I shall quote it below.
Acts 1:4-5 " 4 And while staying with them he charged them not to depart from Jerusalem, but to wait for the promise of the Father, which, he said, “you heard from me, 5 for John baptized with water, but before many days you shall be baptized with the Holy Spirit.”
It is clear that Jesus was not referring to a replacement for Judas in anyway, but to wait in Jerusalem for their baptism of the Holy Spirit; the tongues of fire.
This is delivered in the next chapter, Acts 2:1-4 "When the day of Pentecost had come, they were all together in one place. 2 And suddenly a sound came from heaven like the rush of a mighty wind, and it filled all the house where they were sitting. 3 And there appeared to them tongues as of fire, distributed and resting on each one of them. 4 And they were all filled with the Holy Spirit and began to speak in other tongues, as the Spirit gave them utterance."
The apostles clearly obeyed and listened to Jesus, and waited, as it is not shown in chapter 1 that they ever left Jerusalem.
The Church was not in error in electing Matthias to replace Judas, as it is shown that Matthias was also with them the entire time, as a disciple, Acts 1:20-22 "
20 For it is written in the book of Psalms,
‘Let his habitation become desolate,
and let there be no one to live in it’;
and
‘His office let another take.’
21 So one of the men who have accompanied us during all the time that the Lord Jesus went in and out among us, 22 beginning from the baptism of John until the day when he was taken up from us—one of these men must become with us a witness to his resurrection."
That part of Psalms that Peter is two-fold. The first portion is Psalms 69:24-25. This is in reference to enemies of David, and it is written that after Judas was payed, he bought a farm with that money and "burst open in the middle and his bowls gushed out. And nobody lived there since". The second portion is from Psalms 109:8 which is also about those against the faith, which Judas fulfills, so in fulfillment of the OT, they appointed another to take the place of Judas, another man who had been with them from the baptism to Jesus returning to heaven.
You are arguing that it was Paul that was ment to fulfill this spot, but Paul had not seen or known Jesus during that same time as Judas, and he had not been there from the baptism to Jesus' ascension into Heaven.