There was still a divide between the material world and the spirit world (whatever that looks like to each group). And magic was the work of that spirit realm. Aspirin might be a good example of low magic/alchemy, but seeing the elder for faith medicine was something different.
This divide was nowhere near as firm as you are implying. It was porous as hell (sometimes literally, as in, literally places analogous to the Underworld were porous and ordinary people could visit them simply by finding the physical entrances here on Earth. Consider the myth of Cupid and Psyche.)
I literally did a paper (and presentation) on this in my Latin studies. The Ancient perspective (and to a great extent the Medieval perspective as well) simply, flatly,
did not separate things cleanly along such lines.
Literally the exact same practice could be religious worship, witchcraft, or (natural) philosophy (which is the term that closest corresponds to what we would call "science" today), simply depending on the context in which it occurred. We have recovered curse tablets written by perfectly ordinary Roman citizens hoping the gods would bring about their wishes. Kings and emperors routinely expected things we call "supernatural" to be just...part of the world. E.g. we have correspondence from Grand Duke Cosimo I ordering his advisors to ask their unicorn horn vendor to send a replacement because the one they have
isn't working. And that's from the Renaissance!
So...yeah. This idea that there was the purely supernatural on one side and the purely natural on the other side and they only met in specific ways? Completely a modern invention. To the Ancient and Medieval mind, there was...the world. It had powers in it. Some of those powers were directly perceivable. Others weren't. The riddle of steel was as much "supernatural" as a curse tablet,
especially because it turns out that, coincidentally, using the bones of powerful prey as part of the fuel for the furnace adds trace minerals that do in fact strengthen steel, and may even accidentally create
carbon nanotubes, so some "magical" thinking actually
did produce entirely natural benefits.
It's genuinely very hard to get into the mind of someone from 2000 years ago, not because they were any different from us
cognitively, but because they genuinely believed that gods and spirits and Powers etc. were
also part of the natural world, just an esoteric, complex part. Hence why even (what we might call) "scientific" works would talk heavily about gods and spirits and Powers!
De Rerum Natura ("On the Nature of Things") was, for its time, functionally a "scientific" text, and talks about many things we would call supernatural.