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.
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.