Why do people dying of immune deficiency diseases appear sick?
Many of the symptoms of disease are indeed related to inflammation, but inflammation depends heavily (though not solely) on the innate immune response.  Patients with AIDS and some of the other immunodeficiencies lose their adaptive immune response, not their innate response.  Therefore they are capable of mounting an inflammatory response that is not effective in clearing pathogens (because it doesn't have help from the adaptive immune system) but can still cause symptoms.

More importantly, many symptoms of disease are not caused by the inflammatory response, but are related to organ and tissue damage caused by the infection.  A patient with pneumonia may have a reduced inflammatory response but will still have difficulty breathing and signs of reduced oxygen supply simply because the lung tissue has been damaged by the pathogen.