It is a fairly bleak movie in terms of the role of a folk singer. The movie is about self-defeat and also the consequences of life, but I think the "a ha" moment is that Llewyn realizes what's happening in the end, and that something will change.

It doesn't exactly have a happy ending, but there's a lot of good symbolism between Davis and the cat, Ulysses. There's always someone there to look after it, even if it is a troublemaker and a victim of lots of its own mistakes. In the end, he decides to leave it with its rightful owners, leaving it "how it should be," so-to-speak. This helps wrap up the film in a way. While refreshing my memory on names in the film, I came across an article from The Atlantic which summarizes the end well: "

"He has been awoken from the dream that he’s an undiscovered genius, and from the erroneous notion that talent exists in a vacuum... He still has to pay the price for his behavior... and he still has a long journey ahead. But Llewyn has reconciled with the cat--learning a name that, like Llewyn says of a beloved folk ballad, 'was never new and won’t get old.' And that makes all the difference."

That means that Llewyn has come to terms with his role in life. Sure, it's not a happy ending, but he's come to realize that his folk-singing passion will in the end really be a side-hobby rather than a long-term job. It's a realistic movie about the life of musicians and how they're overlooked by society, sometimes for their faults and sometimes for reasons beyond their control.