His health isn't anyone's business. However, investors have a right to be concerned if a company is too dependent on one person. (This is true even if the person is perfectly healthy; there's always the "hit by a bus" scenario.)

Company executives should be responsible for showing their investors that all critical duties can be handled by at least 2 different people.

Uh, no, it's not. And neither was it people's business whom William Jefferson Clinton elected to sleep with. There is still a right to privacy in this country. And if there is not, then I recommend emigrating to Canada.