I kinda agree with you on all, except this part. 
So we have to go technical for that.
The basic understand for WGS84 is that the datum is processed on the the reciver end, but the datum is maintaind by the DoD. The datum and corresponding parameter loaded into the reciver can be different. You can use the original, the latest or have them all loaded for the user to select which one to use.
The WGS84 datums describes where the origin of the coordinate system starts and more specifically where the core mass of earth (earth gravitational constant) is and this is what we are looking for.
The first original WGS84 constant was a bit of and latest realizations improved this, but this data is not sent with the GPS ephemeris, only the latest updated position of the ground stations are sent via the GPS satellites. The latest WGS84 realization are maintained on reciver level.
So, if you look at this page that you linked to and i quote
The value of GM includes the mass of the Earth’s atmosphere. GPS users should retain the original WGS84 GM value of 3986005.0 108 m3/s2 as specified in the GPS interface control document (ICD-GPS-200) and the NIMA Technical Report 8350.2.
If you look at the Geocentric Gravitational Constant of 3986005.0 (which is the original WGS84) , it correspond to the value in the RTKlib engine (the same used in Reach)
Link to RTKlib manual (se page 143)
and i quote
where:
earth gravitational constant ( 3986005.0 108 m3/s2for GPS and QZSS, 3.986004418 10 m3/s2
for Galileo)
The newest WGS84 realization has a constant of 3986004.418 108 m3/s2
Meaning,
the error you experienced must come from something els.
Maybe this has something to do with it?
https://www.fs.fed.us/GRAIP/downloads/GPS/Understanding%20the%20Evolution%20of%20WGS%201984%20and%20NAD%201983%20(Rev2).pdf
Bottom line, Reach aka RTKlib uses WGS84 the original one, not the latest realization.