I wouldn’t call it a solution, but a workaround - I had to apply an additional step calling gdal_translate to “fix” downloaded raster.
I believe producing transparent TIFFs is a common task among people working with TIFFs, just like specifying GDAL’s nodataValue…
It would be nice to see TIFFs with proper “ColorInterp” flags assigned. If we have a “hack” for PNG, why wouldn’t we have a properly tagged TIFFs?