Well... As a completely recovered alcoholic, I speak from a certain level of authority, at least how it applies to me. There is little difference between alcoholism and other addictions. I suppose alcoholism was the toughest one to beat, but still manageable. I did it myself, with my own unorthodox plan. Nobody believed I could, but I did. And there were times, I came close to failing. Today, nearly a decade later, I can have a few drinks and stop. I'm completely cured.

Does that mean there's no psychological aspect to it? No. Some people are more susceptible to addiction than others. I am one of those people. It's a challenge in life. Does that mean it's a disease? No. Not any more than bipolar, PTSD, or ADHD are diseases. Some claim they are. They aren't. They're simply names we have for sets of behaviours and emotional conditions, but they are demonstrably not diseases. Diseases come from pathogens, and there are no pathogens that cause most "mental illness" or any addictions.

To treat mental conditions, including alcoholism, as a disease is damaging to the patient, and counterproductive. You might argue that the point is so people will treat those suffering with respect, but... As Mitch Hedberg famously put it, "It's the only disease you can get yelled at for having." Treating emotional conditions as diseases when it's convenient and as not diseases when it's not convenient... That's a serious problem in our society.

TL;DR Version: It's more complicated than you're trying to make it.

EDIT: I stopped watching South Park when Cartman killed a kid's parents. That episode was disturbing, and should have had the show cancelled.