I can't repeat your problem. In the beginning I took an png with alpha as reported by gdalinfo
gdalinfo alphatest.png
Driver: PNG/Portable Network Graphics
Files: alphatest.png
alphatest.png.aux.xml
Size is 58, 40
Coordinate System is `'
Origin = (12.139578206197234,40.531369883218098)
Pixel Size = (0.049940902207883,-0.049940902207883)
Image Structure Metadata:
INTERLEAVE=PIXEL
Corner Coordinates:
Upper Left ( 12.1395782, 40.5313699)
Lower Left ( 12.1395782, 38.5337338)
Upper Right ( 15.0361505, 40.5313699)
Lower Right ( 15.0361505, 38.5337338)
Center ( 13.5878644, 39.5325518)
Band 1 Block=58x1 Type=Byte, ColorInterp=Red
Mask Flags: PER_DATASET ALPHA
Band 2 Block=58x1 Type=Byte, ColorInterp=Green
Mask Flags: PER_DATASET ALPHA
Band 3 Block=58x1 Type=Byte, ColorInterp=Blue
Mask Flags: PER_DATASET ALPHA
Band 4 Block=58x1 Type=Byte, ColorInterp=Alpha
I re-used your command
gdal_translate -of GTiff -gcp 567.344 496.649 4.00783e+06 4.42978e+06 -gcp 391.051 1127.79 3.89327e+06 3.44496e+06 -gcp 1595.33 290.84 5.60473e+06 4.91106e+06 -gcp 164.791 32.1259 3.23639e+06 5.03231e+06 -gcp 1562.15 1131.46 5.57244e+06 3.53e+06 -gcp 447.23 586.995 3.84236e+06 4.2981e+06 -gcp 1.73364 49.8788 3e+06 4.936e+06 -gcp 999.31 38.3279 4.5828e+06 5.2809e+06 -co ALPHA=YES alphatest.png alphatest.tif
Gdalinfo proves that ground control points and alpha band are included
gdalinfo alphatest.tif
Driver: GTiff/GeoTIFF
Files: alphatest.tif
Size is 58, 40
Coordinate System is `'
GCP Projection =
GCP[ 0]: Id=1, Info=
(567.344,496.649) -> (4007830,4429780,0)
GCP[ 1]: Id=2, Info=
(391.051,1127.79) -> (3893270,3444960,0)
GCP[ 2]: Id=3, Info=
(1595.33,290.84) -> (5604730,4911060,0)
GCP[ 3]: Id=4, Info=
(164.791,32.1259) -> (3236390,5032310,0)
GCP[ 4]: Id=5, Info=
(1562.15,1131.46) -> (5572440,3530000,0)
GCP[ 5]: Id=6, Info=
(447.23,586.995) -> (3842360,4298100,0)
GCP[ 6]: Id=7, Info=
(1.73364,49.8788) -> (3000000,4936000,0)
GCP[ 7]: Id=8, Info=
(999.31,38.3279) -> (4582800,5280900,0)
Image Structure Metadata:
INTERLEAVE=PIXEL
Corner Coordinates:
Upper Left ( 0.0, 0.0)
Lower Left ( 0.0, 40.0)
Upper Right ( 58.0, 0.0)
Lower Right ( 58.0, 40.0)
Center ( 29.0, 20.0)
Band 1 Block=58x35 Type=Byte, ColorInterp=Red
Mask Flags: PER_DATASET ALPHA
Band 2 Block=58x35 Type=Byte, ColorInterp=Green
Mask Flags: PER_DATASET ALPHA
Band 3 Block=58x35 Type=Byte, ColorInterp=Blue
Mask Flags: PER_DATASET ALPHA
Band 4 Block=58x35 Type=Byte, ColorInterp=Alpha
Parameter -co ALPHA=YES
was not needed but alpha band was created even without it.
I did my test with GDAL 2.4.0dev.