Paul wasn’t depending on the Gospels for his historic account of the resurrection. He depended on eye witnesses. In 1 Cor. 15 Paul dealt with those in Corinth denying a bodily resurrection. That is why he referred to eye witnesses testimonies of the resurrection.
Paul’s purpose in appealing to witnesses still alive is to invite his readers to check his facts if they doubt his words.
Keener, C. S. (1993). The IVP Bible background commentary: New Testament (1 Co 15:6). Downers Grove, IL: InterVarsity Press.
He also noted how critical the resurrection is to the gospel.
For I delivered to you as of first importance what I also received: that Christ died for our sins in accordance with the Scriptures, that he was buried, that he was raised on the third day in accordance with the Scriptures, and that he appeared to Cephas, then to the twelve. Then he appeared to more than five hundred brothers at one time, most of whom are still alive, though some have fallen asleep. Then he appeared to James, then to all the apostles. Last of all, as to one untimely born, he appeared also to me. (1 Cor. 15:3–8, ESV)
Now if Christ is proclaimed as raised from the dead, how can some of you say that there is no resurrection of the dead? But if there is no resurrection of the dead, then not even Christ has been raised. And if Christ has not been raised, then our preaching is in vain and your faith is in vain. (1 Cor. 15:12–14, ESV)
As far as Paul’s account correlating with the Gospels, even the accounts of Christ’s resurrection in the Gospels seem to be contradictory because of the huge amount of detail left out, as John wrote:
Now Jesus did many other signs in the presence of the disciples, which are not written in this book; (John 20:30, ESV).
Now there are also many other things that Jesus did. Were every one of them to be written, I suppose that the world itself could not contain the books that would be written. (John 21:25, ESV)
However, a careful examination of the resurrection accounts reveals they can be consistent if one considers the left out details. This passage in Luke might be a reference to Christ appearing to Peter:
saying, “The Lord has risen indeed, and has appeared to Simon!” (Luke 24:34, ESV)
The appearance to Cephas, that is, Peter, is not recorded in the Gospels other than a passing reference in Luke 24.
Witherington, B., III. (1995). Conflict and Community in Corinth: A Socio-Rhetorical Commentary on 1 and 2 Corinthians (p. 300). Grand Rapids, MI: Wm. B. Eerdmans Publishing Co.
Note that writing material (papyrus and parchment) was expensive at that time and the codex (current book form) wasn’t yet known. Scrolls had a practical size limitation. Thus, the Gospels had less detail than we would like. John’s Gospel is purposely different because he avoided repeating the details in the other Gospels to have room for important details left out of the other Gospels.
Paul gave information about Peter not in the Gospels. He did have direct contact with Peter (Acts 15:7; Galatians 2:11-14).