> The first non-opoid [sic] painkiller
might be:
> The first non-opioid nociceptive pain-killer
Nociceptive pain being that by actual damage to tissue, as opposed to neuropathic pain like a headache or inflammation that you might take a (non-opioid!) NSAID for.
Inflammation would be an instance of nociceptive pain, and an NSAID would alleviate it by reducing the inflammation; and most tissue damage will result in inflammation.
Neuropathic pain, on the other hand, would be due to damage of the nerves themselves, and NSAIDs are completely useless here (ask anyone with sciatica or other nerve entrapment)
Perhaps non-inflammatory (or generically) nociceptive pain killing is the point. (Which is getting a bit specific for such a broad title isn't it. Does 'painkiller' ordinarily have such a narrow meaning in your experience?)
The point isn't to accurately describe it as 'nociceptive' (which I've never heard of before either) and have that be understood, it's to qualify it such that it isn't confusing or sounding like an historical tale of how whatever the first NSAID/analgesic was came to be.