Hollywood has a long history of entanglement with the CIA and DoD in general. (How do you think so many war movies get access to equipment?) The Post was produced by Stephen Spielberg, an entertainer, not a typical careful documentarian. The social purpose of the movie is to soothe liberals at a time when their minds are being fractured by DJT and cries of "fake news". I wouldn't put it past Hollywood and the military to take this prime opportunity to insert propaganda via framing and selection of quotes.

Of course, I can't prove it that this is how this quote came to be, but what I can with certainty is to be skeptical of taking away core moral and factual positions from mass political entertainment and be doubly skeptical of claims (referring to the original claim sans movie) that come from the intelligence services (whose job it is to lie cheat and steal, duh).

EDIT: Here's an example of the CIA getting involved in the movie business https://www.salon.com/2015/09/11/the_cias_insidious_hollywoo...