There is no such thing as "normal exposure" to asbestos.
It is interesting that the lawsuits appear to have been filed by workers at the site, after the great majority of the dust had settled. The people running down the street on 9/11, who were presumably exposed to very high levels, were just bystanders. Certainly the morgue worker in the story would not have been exposed to truly massive levels, nothing like that of workers in shipyards during WWII, for instance.
Interestingly, one explanation for the quick collapse of the towers is that, as I understand it, one of the towers contained no asbestos fire protection material, and the other switched to another material part way up. It is generally agreed that asbestos is the most effective way to protect structural steel against high temperatures.
IOW, the towers quite likely might have stood longer and more people have escaped if asbestos had been used throughout.
>>>Certainly the morgue worker in the story would not have been exposed to truly massive levels,
What? She worked at the site too.