So then the question is, do you go out into the general population with law enforcement, and force people who were drawn in the testing lotto to be tested at gunpoint if they don't comply? I think the answer is that we don't have the will to do that. And perhaps even if this would be incredibly valuable data, it would be morally wrong to gather it in the way we'd have to gather it to have a truly random sample without such a confounding variable.
I am not a doctor, not an epidemiologist. Just some random thoughts.