I use the classical method (Add vector layer, directory, gdb database)
That's the classic way to add vector data. If you want to add raster data, use the Add raster layer method. Or just drag it into your map from the Browser:
If that doesn't work, re-check your GDAL version in QGIS, just in case the GDAL version QGIS is using is different to the GDAL/OGR 3.8.4 you note in your question as raster support was only added to the GDAL OpenFileGDB driver in GDAL version 3.7.
To check:
In QGIS, click Help -> About
So for the above QGIS example, only the 2nd QGIS install would be able to read FGDB raster data.