Thank you for pointing me in the next direction.
Where indeed did these "NT" writers get their material and ideas? I bow to superior minds who posit a source they name "Q", but there is enough speculation without resorting to that.
What were those writers' sources? They were writing in detail of events that took place decades earlier -- often contradicting one another -- or in the case of apocalypses, they write in detail of future events.
I do not think these ideas came to the NT writers "out of the blue". I have material that indicates the sources they used, and which were part of contemporary Judaism's DNA, is to be found in the books of the Second Temple Period, particularly 1 Enoch, Jubilees and the DSS. The NT writers quote from and allude extensively to those writings.
One interesting book is: "Reading Romans in Context: Paul and Second Temple Judaism", Ben Blackwell (etc.) There is more that needs to be investigated.
I wonder if it matters whether a saying attributed to Jesus is "true" or not. I think that what matters is to ask: What was the writer saying to his/their immediate community, and the reason? What was going on at the time in that community at the time they were writing?
This not unreasonable. Read any modern-day polemic and it is pitched at a modern-day community. These modern-day polemics are not being written to some community living in 2000 years time.