Having a similar background (PhD in math, working in deep learning), so knowing nothing of the specifics of medicine, I think that, if anything, there the situation is probably _worse_. Results are less transparent, and you can't just read a paper and know if the results are made "look nicer", if you don't have the raw numbers of the experiments (which you almost never have). At least, in ML you typically can get an idea by reading the paper attentively, and if you're determinate enough, often you even have the source code to check.
I guess that's why in medicine (and some other sciences), meta-reviews are so important: they read dozens or hundreds of papers on the same topic, compare methodologies, and try to deduce more realistic confidence intervals to limit statistical noise. That's how we now know that smoking definitely leads to higher probability of cancer, and most likely that's the case with red meat (and even more likely with smoked or otherwise preserved meat). The risk increases are small enough that a single study can never have the statistical power to prove it. Otherwise, when the famously wrong paper linking vaccines and autism came out, all families of poor autistic children could sue a lot (it was later proved that that one paper was fabricated with malice, but that doesn't change anything, if it were a random error the effect would be the same)
This is why I share your concern for the billions of $ in sues due to this one paper. I have no idea who is abstractly "right" - it might be there was asbestos, or maybe not, and it might be that these talc products increased cancer likelihood, or maybe not. But health, and particularly cancer, is tricky: we all develop multiple cancers, sooner or later, unless we die before. Trying to pinpoint a specific cancer to a source is like fining the proverbial last straw that broke the camel's back - except that in this case it's not the sum of the straws that kill the camel, but rather one random straw across all its load.
That said, I don't want to use this argument to wave company responsibilities away - of course companies who knowingly neglected safety measures, or endangered patient safety, should be punished with large fines (and possibly forced to shut down, in particularly extreme cases). As a European, I'm just uncomfortable with the idea of settling this in spectacular "patient vs. company" trials, where everything is dictated by the ability of lawyers and the will of judges, with results rather randomly ranging from nothing to literally making you rich (and again, your particular cancer might or might not be caused by that... and you can't really know, in any way). With the small additional caveat that you must have survived that in the first place, in order for the company to actually be significantly punished.