An ac system with dirty humidifier portions or stagnant water from the condensate system being plugged can start growing things like mold in the wet filthy areas and cause legionnaires disease or other health issues.

A properly functioning ac system just cools and dries the air.

There is a belief that going from hot and humid summer air to cold and dry ac air and back, constantly all day long, can make you sick.

I have no idea if that is true.

A central hvac system could potentially move particles of infectious materials from one room to another but whether that would make someone sick is debatable due to how short lived virus particles are and other factors.

Some commercial systems utilize ultra violet sections and hepa filters to kill and stop airborne contaminants and allergens.