Articles on these "rationalist" sites often tend towards a mix of smug superiority about one's ability to analyze other people's supposed biases, and lots of references to pet concepts. I posted this one because it does such an excellent job of analyzing the issue in clear, objective terms.
I think the article you linked is an excellent example of smug superiority with lots of references to pet concepts while deficient in regard to the core subjects under discussion. The author demonstrates little practical experience with or research into the actual science around obesity, the actual medical practices related to obese patients, the reality of being an obese person, or the linguistic meaning of the word disease.
It seems you haven't read much else there, and have missed the point; it's not about the scientific details of obesity. And what exactly would be the "linguistic meaning" of the word disease?
I think I've missed the point because there isn't really a point.
The author talks about a hypothetical Doctor but doesn't really seem to know much about medicine. The author asks "what is disease" but doesn't bother to do any real work to answer that question.
To answer your second question, a linguistic analysis of the word disease would look vaguely like the author's "What is a disease?" section, except it would involve real observation about how the word is used and in what contexts. The result is that rather than throwing your hands up in the air saying "it's meaningless to talk about whether something deserves to be a disease" you can say something like: "In a medical context, disease means this' in a research context, disease means this, in a casual context, there are two broad groups of people for whom disease means two different things" Armed with this more accurate language, you can start talking about actual concrete examples and explore more detailed and relevant hypothetical examples.
I think the article you linked is an excellent example of smug superiority with lots of references to pet concepts while deficient in regard to the core subjects under discussion. The author demonstrates little practical experience with or research into the actual science around obesity, the actual medical practices related to obese patients, the reality of being an obese person, or the linguistic meaning of the word disease.