Paul says differently when he explains what the tradition is that he is teaching.
Galatians 1:11-12
"I want you to know, brothers, that the gospel I preached is not something that man made up. I did not receive it from any man, nor was I taught it; rather, I received it by revelation from Jesus Christ."
Paul's letters were written before any Gospels were written.
When he refers to the Gospel, he is referring to the good news from God. He is not referring to Scripture at all, nor even to any of his own epistles, because Galatians is the oldest book of the New Testament.
"I want you to know, brothers, that the gospel I preached"
Where does he say that every bit of the gospel he preached is set down in his surviving writings?