I appreciate your comments in general, but I feel that this point needs to be tempered. Yes there are corrections due to SR and GR effects in GNSS, but they tend to get lost in the morass of the least squares residuals compared to all the other much bigger effects, like clock errors, orbital errors, ionosphere delays, etc, etc.
http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps....
"There is an interesting story about this frequency offset. At the time of launch of the NTS-2 satellite (23 June 1977), which contained the first Cesium atomic clock to be placed in orbit, it was recognized that orbiting clocks would require a relativistic correction, but there was uncertainty as to its magnitude as well as its sign. Indeed, there were some who doubted that relativistic effects were truths that would need to be incorporated [5]! A frequency synthesizer was built into the satellite clock system so that after launch, if in fact the rate of the clock in its final orbit was that predicted by general relativity, then the synthesizer could be turned on, bringing the clock to the coordinate rate necessary for operation. After the Cesium clock was turned on in NTS-2, it was operated for about 20 days to measure its clock rate before turning on the synthesizer [11]. The frequency measured during that interval was +442.5 parts in 10^12 compared to clocks on the ground, while general relativity predicted +446.5 parts in 10^12. The difference was well within the accuracy capabilities of the orbiting clock. This then gave about a 1% verification of the combined second-order Doppler and gravitational frequency shift effects for a clock at 4.2 earth radii."
It makes sense that offset corrections can be handled by a lumped model, as any relativistic corrections on timescales longer than an orbit will be constants of the satellite orbit. Making sub-orbit-duration corrections might require a model that captures more of the relevant physics.