The CIA only locks up brown people and tortures them in Thailand (star). It refers US leakers to the FBI which throws them into lonely prisons, terrifies, and abuses them.
EDIT: (star) Or rather it did. That's what Gina Haspel, our new CIA director nominee, is famous for.
EDIT2: https://www.huffingtonpost.com/norman-solomon/why-the-washin...
A conflict of interest would be relevant, in different ways, if it was current with the events that the movie is about, or if the movie was produced by the Post.
But neither of those is true, sonit seems to be a non-sequitur.
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...
As an example there's a short discussion in the recent movie The Post where Post reporters are trying to figure out who leaked the Pentagon Papers. Ben Bagdikian says something like these guys are always a bit of a showboat and it leads him to remember Ellsberg at Rand.
EDIT to add: the CIA is interested in why people leak because the #1 job of the CIA is to create leakers in other countries.
You mention locking people up, but that is not the CIA's job. The FBI is the counter-intelligence lead in the U.S.