Katherine, this is a very good point to bring up. To me, both from reading the short story and seeing the film, their poverty and lack of education as given facts, not as the central plot, is one of the main aspects of this film that make the characters SO real, and therefore make me feel like I know them, they are like friends. I love them. To me, they HAVE existed, do exist - I will see Ennis, no doubt, when I go to Wyoming!
Also, I am a teacher, and became one in the first place because I believed in education to help kids - especially from poorer backgrounds - have better opportunities in life, and I'm not talking only of the financial aspect. However, even as a teacher, I have never given a fiddler's fart about people's level of education, i.e the number, or names, of degrees on their resumes, and have a definite soft spot for those who could not get an education, particularly for reasons like Ennis's (lack of money).
But you are right, I cannot think of any other film where poverty/class is not central to the plot.