Well how specific is enough to say we do or don’t know how a drug works?
In particular we do know that ASA and other NSAIDS work by inhibiting the activity of the enzyme called cyclooxygenase which leads to the formation of prostaglandins that cause inflammation, swelling, pain and fever. It blocks both COX 1 and 2, though only COX 2 is responsible inflammation. Furthermore, The antithrombotic action of aspirin is due to inhibition of platelet function by acetylation of the platelet cyclooxygenase at the functionally important amino acid serine529.
Now contrast ASA with Acetaminophen …
We know that Acetaminophen also inhibits COX, but only in the CNS and not peripherally. Also, it is only thought that it potentially blocks pain signals via the serotonergic pain pathway.
I would say we know a hell of a lot about aspirin … Acetaminophen not so much on the MOA side of things, however it has been studied so much that we know the safety/toxicity profile like the back of our hands. Either way probably not the best 2 examples to use for your argument.