Did you know that withdrawal from alcohol can kill someone outright? That's probably what killed Amy Winehouse, actually. The risk of death rises each time you relapse and quit. She probably died of repeatedly attempting sobriety, believe it or not.
Did you know that getting clean often leaves someone with permanent mental health problems, like depression, because of what long-term substance abuse has changed in their brain?
In addiction, the brain is re-wired to believe it needs the substance to live. That's why people get sick when they withdrawal, and with a couple substances like alcohol, can even die. It's why they don't necessarily ever return to normal, no matter how long they're clean.
It doesn't mean it's not worth getting clean. Of course it is, and there are endless benefits to quality of life of doing so. But the damage is permanent, and quitting is dangerous, and usually professional medical support needs to be involved. Sometimes forever.
That isn't so much the case with getting back into shape. Assuming you're not attempting to do so through disordered eating, you reap nothing but benefits. It doesn't cause depression. It cures it. It doesn't cause illness. It cures it.
Obese people don't continue to over-eat because they feel like they'll die if they don't. Most of them are actually unable to recognize what hunger feels like, because they never go long enough to actually get hungry. Obesity is, at most, an outlet for something else, which may be a disease (often a mental one). But it simply doesn't have the factors involved that addiction does.
This doesn't clearly answer the question of whether they should get benefits. I for one think it's insane to put health care on employers to begin with and it's unbelievable that we're still trying to run our system this way. But it's not debatable that addiction is a disease. It is often genetic, and it has real physical manifestations in the brain and body.