Yes we do, it's a cyclooxygenase inhibitor. E.g. from the abstract of [1]:

"The mode of action of paracetamol has been uncertain, but it is now generally accepted that it inhibits COX-1 and COX-2 through metabolism by the peroxidase function of these isoenzymes."

Furthermore, the active ingredient in Tylenol is a very simple molecule from a synthetic perspective, and it was developed while trying to overcome the toxicity of an even simpler molecule called acetanilide. Even though it was developed 150 years ago, there was a systematic understanding of organic chemistry and the interaction with the human body already at that time.

[1] https://link.springer.com/article/10.1007/s10787-013-0172-x

I mean, there's plenty of other meds whose exact precise functioning we don't know, just what results and side effects it has.

When tylenol was invented, did the inventors have to precisely specify the molecular interactions between the compound and the human body in the same way a programmer has to precisely specify a program?

No? Maybe the analogy you are suggesting isn’t the most accurate.