I think addiction is a fine word by itself, since it establishes that their behavior is now biological/compulsive, but doesn't group it alongside afflictions that people have far less control over. Disease, even if technically accurate, carries a lot of connotations with it.
If I understand the gist of what you mean, I think disease is still a preferable way of framing the issue.
Two levels of calling addiction a disease come to mind. The first is a technical level, where I essentially defer to my understanding the general medical consensus (though by no means universal). I do believe there are more people that would disagree on this level than you think, but it sounds like we basically agree here. The second is on which is a more effective semantic weapon, which one actually frames the issue better for treatment.
I think, generally, the basic disagreements are going to be over two sets of qualities: size of two pools of people and how far along a spectrum of understanding addiction as a "disease" society [generally] is.
The two pools (very broadly defined):
A) Those that would benefit from a tougher stand against their addiction, other's sympathy might actually enable their habits. This group may benefit from an understanding of addiction as a matter of primarily personal willpower. Such as understanding it less as a "medical" problem (which can feel like it's distant and skewed toward expertise) and finding that they have all the tools they need to control the addiction themselves. Those that find that the better, longer term solution is from an individual place.
B) Those that would benefit from from a more empathetic approach, a more rehabilitative approach perhaps. This group may benefit from an understanding addiction as a "medical" issue as this approach gives more incentive to seeking out advice and expertise. Those that find the better, longer term solution is from a social place.
I'm sure you can guess that I think B is less served than A.
First, I believe neither A or B contain an insubstantial number of people. So I also believe that both approaches have their place contextually.
I think the A is the natural position for most people. I mean, I even often attempt to tough out the non-controversially medical issues more than I should. But it seems to me people are often a tiny bit embarrassed to seek out help when they could use it. And more than a tiny bit when there's an understanding of an issue where it's a personal failing rather than a medical one.
Understanding it less as a personal failing also serves as a preventative measure, in my opinion. Personally, I'd be more likely to heed the effects of addiction if it was considered a physical failing rather than if it was considered a mental failing. I (wrongly) think that I am impervious to tests of will.
Basically, I think addicts are more encouraged to seek assistance if they understand addiction as a disease, and I think more people would benefit from seeking that assistance than not.
To be totally honest, my interest in this area is less about this area and more about the somewhat morally adjacent area of how to treat prisoners.