That page constructs its own argument and then deconstructs it. It doesn't actually disprove the OP's statement.
The reason that GR and SR have "no bearing" is because they've already been designed into the system...
From "Understanding the NAVSTAR" 2nd Edition, by Tom Logsdon ( one of the designers of the system ) the time-dilation is compensated through:
1. Off-setting the clock ticking-rate during manufacture of each satellite.
2. Applying a unique onboard corrective factor according to the eccentricity ( egg-shapedness ) of each satellite's current orbit. The latter is correction is constantly recalculated for each satellite.
Without these corrections against relativistic effects the accuracy would suffer by 14 nm after 24 hours ( without tick-offsetting ) and 100 feet or so per day due to orbital
eccentricity.