It's pretty clear that active addiction, and the following period of post-acute withdrawal that can last a couple years, is a mental health problem, which boils down to a physical medical problem. That doesn't make it a disease, statements from ill-informed doctors from the 1930's (when little was known and the only organized way out was AA) notwithstanding - but, there's abundant evidence that past a point, it's not as easy as simply deciding to quit.
But it does require at least that, deciding to quit, so ultimately it's a choice to decide to quit and stick to it, however you manage that. Hence, it's a choice, as was drinking in the first place, though addictive drinking is perhaps less of a choice. At least, I found myself in that place that I never wanted or intended to go to, before I understood what was happening, but after it was at all easy to turn it around.