I don't think that this is about facing death, I think it's about avoiding life. Ben can't face life and he also can't face death. In the end he fades away. Looking at the protagonist arc from the point of view of a journey from disunity to unity, perhaps Ben's method of suicide by drinking brings him to a form of unity at the very end--because his avoidance of life reaches its inexorable conclusion. Sera sees his core humanity and finds something to love, but I don't think Ben ever really lets her acceptance of him in. In fact, he is brutal in his emotional abuse of her. Perhaps, in looking at dark themes, inventive forms of suicide might be a good topic of discussion. Suicide by cop, suicide by drug lord, suicide by extreme sport, suicide by gambling addiction...

Cage's character seems to me to be actively pursuing death. A slight difference from avoiding life. He is passionate about his pursuit of death. And nothing, not even his passion for Sera, will dissuade him from his path. Sera, on her part, accepts his condition: if you want to love me you will never ask me to give up drinking/dying. She understands his level of despair and will not abandon him, this devotion defines their love and in a way defeats death.

What you see as actively pursuing death is what I see as actively avoiding life. I think it's Ben's pursuit of escape that is passionate. I think that a person who passionately wants to die would does not linger while suffering all the pain of advanced liver disease and DTs. There are easier ways to go. Another way of putting it might be that Ben is tragically conflicted. He dances down the liquor aisle while filling his cart with bottles of booze, but does that mean he's sincerely happy? It's a great movie, and superbly acted. I think that many people have seen someone who has chosen a similar path.