How do we explain Paul's encounters with disciples of Jesus in light of the Great Commission? The following passage relates the "Great Commission" that Jesus gave to 11 disciples:

https://www.biblegateway.com/passage/?search=Matthew%2028:16-20

But Paul, a man who never met Jesus during his life, seems to have taken up the mantle all by himself:


  13I am talking to you Gentiles. Inasmuch as I am the apostle to the
  Gentiles, I take pride in my ministry 14in the hope that I may somehow
  arouse my own people to envy and save some of them. 15For if their
  rejection brought reconciliation to the world, what will their
  acceptance be but life from the dead? 16If the part of the dough
  offered as firstfruits is holy, then the whole batch is holy; if the
  root is holy, so are the branches.


In Acts 10, Cornelius comes to Peter, and he is already following God. And then Peter baptizes him. OK, but that's hardly "going out and making disciples of all the nations".

In Acts 15, Peter seems to recall that he (alone?) was supposed to be apostle to the Gentiles. He concludes by saying:


  19“It is my judgment, therefore, that we should not make it difficult
  for the Gentiles who are turning to God. 20Instead we should write to
  them, telling them to abstain from food polluted by idols, from sexual
  immorality, from the meat of strangled animals and from blood. 21For
  the law of Moses has been preached in every city from the earliest
  times and is read in the synagogues on every Sabbath.”


So it was only then decided what the gentiles should do. Which means that this whole time, it was mostly Paul converting the gentiles. What happened to the 11 apostles? Where are their epistles? Who were they converting among the gentiles? According to the Great Commission, they should be baptizing them in the name of the trinity. Why was Paul given the job?

Is part of the explanation that the Great Commission was added later to the Gospel of Matthew?
Throughout the Gospels, the apostles at times seem absurdly hard-headed and unable to understand what Jesus is telling them. This only continues in Acts, and it seems that the author wants readers to think the apostles slow to understand. But the Holy Spirit had his own ideas for the church, and the apostles were blown along. I don't think the argument about Matthew's ending being added later is valid since Luke's ending is similar: Jesus says, "Thus it stands written that the Christ would suffer and would rise from the dead on the third day, and repentance for the forgiveness of sins would be proclaimed in his name to all nations, beginning from Jerusalem." (Luke 24:46-47)

Then in Acts 2, the disciples are gathered in Jerusalem for the feast of Pentecost. "Now there were devout Jews from every nation under heaven residing in Jerusalem." (Acts 2:5) People from all nations are there, but they all happen to be Jews! Can you see why it may not have occurred to the disciples yet that the Gospel is to be preached to Gentiles? But God wasn't done. He brought a Gentile, Cornelius, to Peter in Acts 10. It was puzzling at first, but you can almost see the light-bulb above Peter's head as he describes his acceptance of Cornelius' conversion (Acts 10:34-39,47):


  I now truly understand that God does not show favoritism in dealing with people, but in every nation the person who fears him and does what is right is welcomed before him. You know the message he sent to the people of Israel, proclaiming the good news of peace through Jesus Christ (he is Lord of all)— you know what happened throughout Judea, beginning from Galilee after the baptism that John announced: with respect to Jesus from Nazareth, that God anointed him with the Holy Spirit and with power. He went around doing good and healing all who were oppressed by the devil, because God was with him. We are witnesses of all the things he did both in Judea and in Jerusalem. ... No one can withhold the water for these people to be baptized, who have received the Holy Spirit just as we did, can he?


It's to this incident that Peter refers in Acts 15:7-9 when he says, "God chose me to preach to the Gentiles so they would hear the message of the gospel and believe. And God, who knows the heart, has testified to them by giving them the Holy Spirit just as he did to us, and he made no distinction between them and us, cleansing their hearts by faith."

Paul reports that the apostles accepted him as one of their rank (Galatians 2:7-10; also reported by Luke in Acts 15:22-27):


  When they saw that I was entrusted with the gospel to the uncircumcised just as Peter was entrusted with the gospel to the circumcised (for he who empowered Peter for his apostleship to the circumcised also empowered me for my apostleship to the Gentiles) and when James, Cephas, and John, who had a reputation as pillars, recognized the grace that had been given to me, they gave to Barnabas and me the right hand of fellowship, agreeing that we would go to the Gentiles and they to the circumcised. They requested only that we remember the poor, the very thing I also was eager to do.


So yes, it seems that there weren't large-scale missions to the Gentiles until Paul came along. But there was Philip and the eunuch from Ethiopia, Peter and Cornelius, and so on.

The question that then arises (and is implicit already in your original question) is, why didn't the other apostles start preaching to the Gentiles once they realized it was the mission of the church? There are a few angles to this question.

First of all, Paul never went on his missions alone. In his letters he frequently mentions Apollos, Barnabas (who is also a Jew), Mark, Timothy, Titus, and Silas. Mark, incidentally, is also affectionately mentioned by Peter in one of his epistles. This is definitely not even close to an exhaustive list of his fellow laborers.

Secondly, just because we don't have Biblical evidence of the original 12 apostles' missions to the Gentiles, doesn't mean no such missions happened. But most of them do have extensive traditional biographies (hagiographies) that place them in exotic far-away lands preaching the Gospel. Perhaps most famous is Peter himself (becoming bishop of Rome) and Thomas (carrying the Gospel to India). And traditionally John ended up residing in Ephesus. Of course, John and Peter both have NT epistles attributed to them, written to the whole church. So do James and Jude, Jesus' brothers. And John also wrote Revelation, which is replete with passages about the church being comprised of both Jews and Gentiles and people of all nations, tribes, and tongues. So it seems clear that even if Paul was the one primarily responsible for carrying the Gospel to the Gentiles, others were doing it too.

Thirdly, even if the other apostles' primary work was among their own people, what's wrong with that? Usually when a missionary preaches to a certain people, his chief goal is to raise up leaders who can communicate the Gospel to his own people. Africans most effectively preach to Africans, Polynesians to Polynesians, Americans to Americans. Why wouldn't Jews be the most effective instruments of conversion to Jews? Why shouldn't Jesus' Jewish followers aim to convert their fellow Jews? In the same way, when Paul visited various cities, his main goal was to plant a new church whose leaders were from that city. He also affirmed that the Gentiles were "ingrafted" into Israel (Romans 11), and that the Gospel was to go to "the Jew first, then to the Gentile." (Romans 1:16)

So a few points should now be clear: 1) The apostles were slow to understand the nature of God's mission to the Gentiles. 2) The apostles accepted Paul's mission to the Gentiles. 3) The apostles and other believers participated in that mission once they understood it. 4) It is better (more effective) for individuals to witness to their own culture than to another.