No, not aspirin. We know aspirin very well in the medical community.
TBF there are drugs out there that we do not know the MOA of, like methocarbamol (from the national institute of health: “The exact mechanism of action of methocarbamol remains unknown; similarly unknown is the relationship between musculoskeletal pain and muscle spasm” lol)
So for long term safety, it is based on animal and human studies. These studies happen for multiple years prior to being put on the market (for the most part, though that is a story for a different day). Then after the drug is on the market, the drug company is required to do “Postmarket Clinical Studies” to show that their drug is still doing what it was initial shown to do; furthermore, to look for safety events of said drug.
A really famous case of a bungled postmark study was Vioxx. Vioxx is/was a Cox-2 specific pain medication. In the initial and postmarket studies they found that it had an increased risk of heart attack (in some cases up to 88% increased risk). The company Merck held the information from the public and FDA. They were forced to take the drug off the market in 2004. Technically in short bursts Vioxx was probably safe, but long term it was not.