I thought they were called "woodwinds" due to the wooden reed, not the instrument material. Saxophones are brass, but considered woodwinds. I used to play sax in school.
Then again, flutes are considered woodwinds, but there's no wood on them at all now. I assume they perhaps used to be wood, or just from the fife ancestry. But that goes against my reed theory.
Oh well, consistency is overrated anyway! [
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BTW, my mom and dad have clarinets from when they were in school back in the 50s/60s. They're both made of wood - not sure what kind though.