Before I respond to your question, I have to repeat to NEVER use epsg.io! It is an unofficial site that has serious bugs and uses an outdated copy of the database from the official epsg.org.
Thanks for the additional information. This confirms my suspicion that problem is that you do only the projection and not the datum transformation (although the first WKT string does specify outdated approximate datum transformation parameters).
If you use (pyproj with) a uptodate PROJ version and an uptodate EPSG database version you can get the correct WKT string with the projinfo command:
\>projinfo -s epsg:4937 -t epsg:28992``
...
CONCATENATEDOPERATION["Inverse of Amersfoort to ETRS89 (9) + RD New",
SOURCECRS[
GEOGCRS["ETRS89",
ENSEMBLE["European Terrestrial Reference System 1989 ensemble",
MEMBER["European Terrestrial Reference Frame 1989"],
MEMBER["European Terrestrial Reference Frame 1990"],
MEMBER["European Terrestrial Reference Frame 1991"],
MEMBER["European Terrestrial Reference Frame 1992"],
MEMBER["European Terrestrial Reference Frame 1993"],
MEMBER["European Terrestrial Reference Frame 1994"],
MEMBER["European Terrestrial Reference Frame 1996"],
MEMBER["European Terrestrial Reference Frame 1997"],
MEMBER["European Terrestrial Reference Frame 2000"],
MEMBER["European Terrestrial Reference Frame 2005"],
MEMBER["European Terrestrial Reference Frame 2014"],
ELLIPSOID["GRS 1980",6378137,298.257222101,
LENGTHUNIT["metre",1]],
ENSEMBLEACCURACY[0.1]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
CS[ellipsoidal,3],
AXIS["geodetic latitude (Lat)",north,
ORDER[1],
ANGLEUNIT["degree",0.0174532925199433]],
AXIS["geodetic longitude (Lon)",east,
ORDER[2],
ANGLEUNIT["degree",0.0174532925199433]],
AXIS["ellipsoidal height (h)",up,
ORDER[3],
LENGTHUNIT["metre",1]],
ID["EPSG",4937]]],
TARGETCRS[
PROJCRS["Amersfoort / RD New",
BASEGEOGCRS["Amersfoort",
DATUM["Amersfoort",
ELLIPSOID["Bessel 1841",6377397.155,299.1528128,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4289]],
CONVERSION["RD New",
METHOD["Oblique Stereographic",
ID["EPSG",9809]],
PARAMETER["Latitude of natural origin",52.1561605555556,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",5.38763888888889,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",0.9999079,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",155000,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",463000,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["easting (X)",east,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["northing (Y)",north,
ORDER[2],
LENGTHUNIT["metre",1]],
ID["EPSG",28992]]],
STEP[
COORDINATEOPERATION["Inverse of Amersfoort to ETRS89 (9)",
SOURCECRS[
GEOGCRS["ETRS89",
ENSEMBLE["European Terrestrial Reference System 1989 ensemble",
MEMBER["European Terrestrial Reference Frame 1989"],
MEMBER["European Terrestrial Reference Frame 1990"],
MEMBER["European Terrestrial Reference Frame 1991"],
MEMBER["European Terrestrial Reference Frame 1992"],
MEMBER["European Terrestrial Reference Frame 1993"],
MEMBER["European Terrestrial Reference Frame 1994"],
MEMBER["European Terrestrial Reference Frame 1996"],
MEMBER["European Terrestrial Reference Frame 1997"],
MEMBER["European Terrestrial Reference Frame 2000"],
MEMBER["European Terrestrial Reference Frame 2005"],
MEMBER["European Terrestrial Reference Frame 2014"],
ELLIPSOID["GRS 1980",6378137,298.257222101,
LENGTHUNIT["metre",1]],
ENSEMBLEACCURACY[0.1]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
CS[ellipsoidal,3],
AXIS["geodetic latitude (Lat)",north,
ORDER[1],
ANGLEUNIT["degree",0.0174532925199433]],
AXIS["geodetic longitude (Lon)",east,
ORDER[2],
ANGLEUNIT["degree",0.0174532925199433]],
AXIS["ellipsoidal height (h)",up,
ORDER[3],
LENGTHUNIT["metre",1]],
ID["EPSG",4937]]],
TARGETCRS[
GEOGCRS["Amersfoort",
DATUM["Amersfoort",
ELLIPSOID["Bessel 1841",6377397.155,299.1528128,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
CS[ellipsoidal,2],
AXIS["geodetic latitude (Lat)",north,
ORDER[1],
ANGLEUNIT["degree",0.0174532925199433]],
AXIS["geodetic longitude (Lon)",east,
ORDER[2],
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4289]]],
METHOD["Inverse of HORIZONTAL_SHIFT_GTIFF"],
PARAMETERFILE["Latitude and longitude difference file","nl_nsgi_rdtrans2018.tif"],
OPERATIONACCURACY[0.001],
ID["INVERSE(DERIVED_FROM(EPSG))",9282],
REMARK["Replaces Amersfoort to ETRS89 (7) (tfm code 7000). Horizontal component of official transformation RDNAPTRANS(TM)2018."]]],
STEP[
CONVERSION["RD New",
METHOD["Oblique Stereographic",
ID["EPSG",9809]],
PARAMETER["Latitude of natural origin",52.1561605555556,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",5.38763888888889,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",0.9999079,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",155000,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",463000,
LENGTHUNIT["metre",1],
ID["EPSG",8807]],
ID["EPSG",19914]]],
USAGE[
SCOPE["unknown"],
AREA["Netherlands - onshore, including Waddenzee, Dutch Wadden Islands and 12-mile offshore coastal zone."],
BBOX[50.75,3.2,53.7,7.22]]]
...
The grid file nl_nsgi_rdtrans2018.tif
that you need, is included (without the prefix nl_nsgi_
in the name) in the RDNAPTRANS download you already requested.
If you need physical heights in the national height system you need to use EPSG:7415 instead of EPSG:28992.