Yes, it's "obviously true" because of the confounding factors: bad air tends to go hand in hand with dirty water, contaminated food, and infection-carrying insects.
Neither cholera, malaria, typhoid nor plague are airborne, and yet their causes were ascribed to miasmata.
Right but the metabolites produced by the bacteria are harmful in their own right. And let's be clear, there are many airborne contagions like the common cold that haven't been indexed through the centuries. It isn't a good idea to inhale shit air for days on end, just like it isn't a good idea to inhale car fumes for days on end.
Neither cholera, malaria, typhoid nor plague are airborne, and yet their causes were ascribed to miasmata.