Actually, there are a LOT of passages in the New Testament about this. However, this is a teaching that Jesus gave fully to His disciples (including St. Paul) after His death and resurrection. St. Paul, indeed, was instructed by Jesus privately (as he tells us in his epistles) as he (Paul) was specifically chosen to be the apostle to the Gentiles. However, all through the Acts of the Apostles we see that it is written that in the 40 days Jesus was on earth after His death and resurrection, and before His ascension into Heaven, he spoke and explained His teachings to the apostles (who in turn went on to explain both in writing and orally–what we know as Sacred Tradition). . .and furthermore, Jesus sent the Holy Spirit to guide the Church into ‘all truth’. It is all ‘there’ but things needed to be developed from ‘bare bones’ etc. as people reached greater understanding.
If you, Ben, are going to accept as a working premise that the gospels record the words of Jesus (but not all He said and did, scripture is explicit that it does not in fact list every word and deed of Jesus), then you also have to accept that Acts, and the epistles, likewise record apostolic teaching . . .and the apostles are teaching what they received from Jesus.
That means that even if you don’t read in King James (or New American, or whatever) English that “Jesus stood up and said, Remember, I have come to Israel first and then the Gentiles, so queue up”. . . if you do read further on the words of St. Peter and St. Paul in Acts, or Romans, Hebrews, etc., and find in the Catechism of the Catholic Church, etc., this teaching, then this was indeed a teaching Jesus gave but did not necessarily cause to be ‘written down’ in ‘His own words’.
Now if you’re not going to accept that, you have to give a reasonable reason that you’ll accept only something ‘said’ in the gospels of Matthew, Mark, Luke and John, but not in Acts of the Apostles or any epistles.