This is at best a fallacy with occasional exceptions and in this particular case, just plain incorrect.

If it works by 'blocking a compound that supports nerve cells in transmitting pain' it isn't doing anything to correct the pathology. You are, in the truest sense of the way that people naively talk in derogatory terms around modern medicine, 'treating the result, not the cause'

It's definitely not a fallacy that things that relieve pain can also help heal or prevent damage, NSAIDs are a great example. It mentions it may do more than just blocking the signal in the article.