Not every drug is an opioid. We have prohibition laws designed for opioids blindly applied to any (in the western context) nontraditional drug. The German law on drugs is literally called "the painkiller law", for instance.
The meaning has drifted, appropriately enough. Betäubungsmittel originally meant painkillers, as you can tell from the word. It's just that now every recreational drug is labeled as such.
You seem to be confusing the words "Schmerzmittel" (analgesic, pain killer) and "Betäubungsmittel" (narcotic). Those two classes of substances are not the same.
"Betäubung" has a similar etymology as "narcotic". Both mean to numb the senses or put to sleep (hence e.g. "narcolepsy"), and in German it's therefore also used for sedatives and anesthetic drugs. In modern use, "narcotic" has also semantically shifted to include any illegal drug, as with "Betäubungsmiddel".
Interestingly, in both cases the semantic shift seems to have been caused by the enactment of laws to control drugs. The legal term these days is probably "controlled substance" in English, but "narcotic" now definitely refers to many drugs that are not medically narcotic.
It can also mean anesthetics, which coincidentally would include cocaine as a strong local anesthetic, but not a narcotic in the pharmacological sense.
1. https://wetten.overheid.nl/BWBR0001941/2026-01-28