Which is what he addresses in Galatians, especially chapter 1.
Galatians 1:6 I am astonished that you are so quickly deserting the one who called you by the grace of Christ and are turning to a different gospel 7which is really no gospel at all. Evidently some people are throwing you into confusion and are trying to pervert the gospel of Christ. 8But even if we or an angel from heaven should preach a gospel other than the one we preached to you, let him be eternally condemned! 9As we have already said, so now I say again: If anybody is preaching to you a gospel other than what you accepted, let him be eternally condemned!
My guess is that likely the reason he didn't rehash what happened in the Gospels was that it was because what was recorded in the Gospels for us, was pretty common knowledge to them, back in the days when Paul wrote his letters.
And my original response was that the WRITTEN Gospels appeared AFTER Paul was martyred and what was considered "common knowledge" was as likely to be wrong as it was right.