Modern Times (1936) is perhaps one of the best examples of an American film that has strong leftist themes.  The entire film revolves around worker strikes, issues of the Great Depression, and I would even argue: an early critique of American consumerism (The American Dream).

The main character (the famous “Tramp”) goes through various difficulties in life throughout the film, including finding employment, arrest, etc.  Early on in the film, the capitalists in charge of the factory that the Tramp works in attempt to implement a way to manage lunch breaks more “efficiently” by introducing a machine that automatically feeds the worker to reduce the amount of time taken off by the worker for lunch.  This comedic scene highlights a classical problem with Capitalism: that of time management and exploitation.

The Tramp accidentally leads a socialist march

The Tramp also has a love interest in the film who also runs into employment problems throughout the film.  They eventually get a shack to live in that is falling apart, while at one point in the film they both dream of the typical “American Dream” house (that dream sequence seems to also be an appropriate critique of the the conservationism of the 1950s before its own time).

The two also at one point are in a department store enjoying the luxuries of being able to use everything in the store.  That fantasy sequence itself is a critique of consumerism and the constant promotion of consumer goods when workers (especially during those times) had a tough time having access to said goods.

Modern Times is an important critique of Capitalism and is itself a great film.