I am surprised that this thread has gone on for 3 pages. I'm afraid I haven't had the time to go through each and every response. I still have a remark that I hope will be helpful.
My $.02. It would indeed be possible, in theory, to not adjust the rate of the GPS clocks. For a reference, see "Precis of General Relativity" by Misner,
http://arxiv.org/abs/gr-qc/9508043 (or
http://arxiv.org/pdf/gr-qc/9508043v1.pdf for the pdf).
For GPS the measuring instruments can be taken to be either ideal SI atomic clocks in trajectories determined by known forces, or else electromagnetic signals describing the state of the clock that radiate the signal. Each clock maintains its own proper time (but may convert this via software into other information when it transmits).
I believe Misner's paper was written in response to Ashby's.
However, it at least appears CONVENIENT, though not NECESSARY, to have the clocks transmit a coordinate time rather than proper time, specifically the coordinate time called atomic time, also known as TAI time. I was curious, personally, as to which approach the European Union's "Galileo" positioning system used, but I wasn't able to find anything definite in the amount of time I had to dedicate to find what sort of time the Galileo satellites transmitted.
Having the clocks transmit their coordinate time, rather than their proper time, makes analysis via coordinates easier (as long as you've settled on a consistent coordinate choice), and hopefully easy to understand as well. But either approach will do.
It is of course not absolutely NECESSARY to use any particular coordinate system. Usually the position coordinates are desired in an ECEF coordinate system (Earth centered, Earth fixed) - because in this coordinate system, points on the rotating Earth have constant coordinates, which is what's generally meant by a "position" on the rotating Earth. Light rays do not follow straight lines in the ECEF coordinates, however, so the usual choice is use a different coordinate system, ECI (Earth Centered Inertal) coordinates for the bulk of the analysis, which makes analysis of the trajectories of the radio signals easy because the radio signals can be presumed to travel in straight lines in this coordinate system (with the possible exception of some minor atmospheric effects). Then one converts the results from ECI back to ECEF at the end of the analysis. The usual choice for the time coordinate system is known as atomic time aka TAI time, thus the time coordinate used by TAI are hopefully familiar to the reader (though all of the technical details may not be familiar). Note that civil time, UCT, aka Coordinated universal time, is derived from TAI time by the addition of leap seconds to keep UCT in synch with the sun / solar time. TAI time is equal to proper time for objects at rest on the surface of the Earth only at sea level (more technically, on the geoid), as several other posters have already noted.
I also rather strongly suspect that some of the non-science-advisor posters in this thread are clinging to the notion of absolute time, most likely without realizing it, and that this is the ultimate source of their confusion with regards to the issue of time dilation. Unfortunately, I'm not aware of any good strategy to clear up this sort of confusion :(.
If there is interest, and I have the time, I might try to write something up about the relationship between the relativity of simultaneity and time dilation, but experience has shown me that it seems a hard point to get across.