What color interpretation gdalinfo shows for the selected bands 1, 2, and 3? Are they "undefined", or some color like here
Band 2 Block=12000x1 Type=UInt16, ColorInterp=Green
Gdal_translate copies the color interpretation from the source image but it can be overridden with the -colorinterp
option https://gdal.org/programs/gdal_translate.html#cmdoption-gdal_translate-colorinterp. In your case the command would be
gdal_translate -ot UInt16 -b 1 -b 2 -b 3 -r cubic -of PNG -colorinterpr red,green,blue "C:\Users\Cemre\Downloads\0309_0_aug_1.tif" "C:\Users\Cemre\Downloads\0309_0_aug_1.png"
Notice that png format does not support color interpretation and therefore this info is stored into .aux.xml sidecar file
...
<PAMRasterBand band="2">
<ColorInterp>Green</ColorInterp>
...
If a viewer is aware of the meaning of aux.xml files it can apply the correct color interpretation. One example of such viewers is QGIS.