Clarification
Running
gdal_translate in.tiff out.tif -b 1 -b 2 -b 3 -b 4
solves the issue - gdalinfo doesn’t report any warnings, the bands look like:
Band 1 Block=864x2 Type=Byte, ColorInterp=Red
Mask Flags: PER_DATASET ALPHA
Band 2 Block=864x2 Type=Byte, ColorInterp=Green
Mask Flags: PER_DATASET ALPHA
Band 3 Block=864x2 Type=Byte, ColorInterp=Blue
Mask Flags: PER_DATASET ALPHA
Band 4 Block=864x2 Type=Byte, ColorInterp=Alpha