I think you're touching on the heart of the tension behind the "health at every size" movement.
I know a couple of people who are big advocates of it, and surprise surprise, they're all pretty fat. They're still wonderful people, and I of course don't respect them less because they're fat. You know, for me, it's like..who cares? It's not my body. I'm even pretty sympathetic to how much difficulty they have doing basic stuff, just for having their joints and muscles carrying around an extra hundred pounds.
But it really seems to me that what's going on is that they're reacting to the constant onslaught of fat-shaming and moral judgments with this ideology of "health at every size". There is a constant moralizing about fat people in our culture, like if you're fat you're lazy or bad or whatever, none of which is warranted.
But I think that's where things get muddled--culture turns fatness into a moral failing, wrapped under a sheath of health concern trolling. Reaction to the moral judging is to try and take away the health claims, when that's really kind of missing the point.
What would be appropriate, I think, is to push for "human dignity at every size," you know, and leave the moralizing to stuff that matters. But that's not so catchy or easy to distinguish, I guess...
As for the actual health claims, I, for one, don't buy it at all, it seems obvious to me that carrying around excess weight is going to cause a number of immediate problems (joint strain, etc), to say nothing of the strain on your organs for being encased in excess fat. But the health stuff is separate from the moralizing, and I don't know how you'd get the larger culture to separate out "you're fat and it's hurting your body" from "you're fat and you're lazy/stupid/etc".