I'm not sure if it's really fair to expect such movies, which are a tiny minority of all movies made, to have such a disproportionate impact on policy, especially considering that they're usually ignored or even actively despised by the media and the powers that be.
I don't know of specific examples in film, but some books have had such impact. The Jungle[1] and Silent Spring[2] come to mind.
Film is a relatively young media, and its full potential is unlikely to be realized for decades if not centuries. Give it some time. I'm optimistic that truly moving and motivating films will arrive.
[1] - https://en.wikipedia.org/wiki/The_jungle#Federal_response
It's almost as if the artist can only bring out something that already exists in the audience. Everyone sees it through their own personal lens.
If a movie existed that realistically portrayed poverty, that would only move the viewer to the extent that they can relate to the characters. One person might be motivated to reform the social safety net to provide better support. Another might reaffirm their personal preconception that poor people are brutish, unintelligent, useless, and irredeemable, and might just sign on to a new eugenics program to sterilize indigents and undesirables.
The art rekindles empathy that is already there. If it wasn't already present, it is a lot more difficult to make it grow there from other places in the psyche.
"And perhaps if others saw more realistic portrayals of poverty in the movies, they might be more receptive to programs that would help them instead of demeaning and punishing them."
I would love an example where 'realistic portrayals' of a given subject in movies led to changes in public policy / public perception.
If anything, I would argue that popular films, just like other art forms, are more likely to serve as a barometer of society's perception/reception of a given subject than as something that ignites the change.
Take the war in Vietnam in the 60's and 70's - I'm not sure you could make 'The Deer Hunter' in '68, or 'The Green Berets' in '78.
Or the war on drugs - movies like Trainspotting, Requiem for a Dream, or Traffic certainly showed a realistic side of drugs and addiction... and yet the war on drugs has only intensified since their popular releases.