This is clickbait, there are at least two classes of non opioid painkillers in use since a long time.
https://en.wikipedia.org/wiki/Analgesic
The term 'painkiller' is reserved for strong pain relief, and wouldn't include things like ibuprofen. That made me think immediately of a non-opioid pain blocker not just a pain reliever.
Having claimed it was cultural. It have been helpful of you indicated which culture you felt this applied to.
In UK I'd say painkillers includes ibuprofen and paracetamol. I suppose with ibuprofen it's also referred to as an anti-inflammatory. Not sure how else one would refer to paracetamol other than with synonyms (analgesics) or euphemism (pain relief tablet).
I particularly disagree with the parent comment that calls this click bait. The topic's intrinsically interesting to anybody who'd be lured in by that title; it doesn't need "bait" and we all know NSAIDs exist.
The article's particularly good at citing its references inline, which I very much appreciated. Added this author to my RSS reader in fact.
In which dialect of English do you think this is true?
Maybe there's some very specific, limited medical context where this is the case but in common parlance it's not at all the case, search for "painkiller" in an online shop like Amazon and you'll find a whole lot of Paracetamol/Tylenol, and various NSAIDs (aspirin/ibuprofen) and manufacturers of those drugs actually use that specific term.