If there is a narrow view of what "the gospel" is, then that could explain the confusion in the question. There are four gospel accounts, written between the early 50s through till around 85. Paul was martyred in 67/68, so the gospel account of John was never even available to him, and if Luke was written in the 70s, neither was it. However, a gospel account is not the same as "the gospel of Christ". Let me explain.
None of the gospel accounts are biographies of Jesus Christ, but they all have particular emphases on the life, death, and resurrection of the Messiah. They were written for different readers for different purposes. When combined they give a suitably fulsome word-picture about the Christ which confirm who he is and what he said and taught. It was the apostle Paul, however, who wrote in great detail the extent of what encompasses "the gospel of Christ". (See 1 Corinthians 9:12; 2 Corinthians 4:3-5; 2 Thessalonians 1:8.)
After his miraculous conversion, Paul had the gospel about Christ explained to him from Ananias and other Christians in Damascus. About three years passed (mainly spent in Arabia, whose borders extended to the environs of Damascus), then he was taken to meet the apostles in Jerusalem, who were initially afraid of him. But Barnabas introduced him. Again, once Paul got going preaching the gospel to Jews, his life was threatened, so effective was his ability to show from the Hebrew scriptures that Jesus is the Christ (Acts 9:1-30 & Galatians 1:6-24).
His Judaic theological training (second to none) meant he concentrated on quoting from the Hebrew scriptures. His habit was to "preach first to the Jew, then the Gentile" (until the Jews excluded him). See Acts 13:14. That is why his quotations were from the Hebrew scriptures - to prove that Jesus is the Christ. It is also necessary for Gentiles to learn from the Hebrew scriptures all that pertains to the gospel, from the first prophecy about salvation in Genesis 3:15 right through to the last prophecy in Malachi 4:5-6.
By the time Paul's letters began to circulate, some perversions of the gospel had begun to creep into congregations, corrupting the true gospel message. The need of the hour was not to repeat what Matthew, Mark and Luke had already written in their gospel accounts, but to show the application to the newly forming congregations. Indeed, for the last five years of Paul's life, he was prevented from freely moving around the congregations and to preach. He had been arrested in Jerusalem, detained for two years in Caeserea, then sent in chains to Rome. That journey took about two years, then he was held under house-arrest for around two more years. It was about 35 years since Christ's resurrection. A new generation was arising, only the remaining elders remembering having seen Jesus after the flesh. Paul's writing was to counter the false gospels particularly taking root in those congregations he had established. That is why they are theological letters, dealing with the practical outworkings of living the gospel of Christ, based on accurate knowledge of the deity of the risen Christ; it's his resurrection that proves him to be the Son of God, and guarantees the resurrection of every Christian. Paul has to argue from the Hebrew scriptures why Christians must not let themselves be bound again (with legalism); how their freedom in Christ means righteous living, not licentiousness. That sort of thing.
The writings of James and Peter are much shorter than Paul's. James and Peter were 'ordinary' 'unlettered' men whose letters were on specific matters that really did not call for quotes from the gospel accounts. But they do make a few references to Jesus' words and events. See 1 Peter 2:4-8 [a reference to events Jesus spoke of in Mat.21:41-44] 2 Peter 1:12-18; 3:2 & 18; James 1:27 [a reference to Mat.25:36, Jesus' own words] & 3:12 [a reference to Jesus' words in Mat.3:17] & 5:12 [ditto Mat.5:34-37] also Jude vss.17 & 21.
Bear in mind, too, that a letter is not a book. When its frame of reference is very limited, it will be quite short. It simply is not reasonable to alight on them with a call for "justification" as to why they hardly mentioned what you assume to be what they ought to have written about. Yet they DO deal with Jesus' words, and gospel events - just not in a direct word-for-word quotation that you seem to be looking for. All their writings are imbued with this zeal for Jesus' words and belief in gospel events.