The manufactured statements that your Didache puts in the mouth of Paul do not comport with his own words of the letters...read them carefully.
It’s not “my” Didache. LOL!
It was written by the apostles.
If it was either co-written by Baranbas and Paul, or “ a set of lecture notes taken while Barnabas and Paul and Titus were speaking,” it would not necessarily sound like one of Paul’s letters to a church that he had established.
I can say that my notes of my college professor’s lections would not have given the same ‘sense’ as one of their lectures on the same topic that was written out.
And this is more of a report to the council in Jerusalem than a letter to a congregation.
The Didache came from the apostles.