Does GPS Give Experimental Support to Theories of Relativity?

You often hear physicists claiming that the fact that GPS works so well gives experimental evidence that Einstein's theory of relativity with clocks being affected by gravitation and motion, is correct: Stephen Hawking states in his bestseller A Brief History of Time that without relativistic corrections, your GPS would be off by kilometers.

But is this true? The facts are: GPS-satellites carry identical atomic cesium clocks adjusted at launch to 10.22999999543 MHz from the 10.23 MHz of the reference ground clock, which corresponds to 38 nanoseconds per day. Physicists claim that this results from a gravitational effect of general relativity making the satellite clocks tick 45 nanoseconds fast and a velocity effect in the opposite direction by special relativity of 7 nanonseconds giving 38 = 45 -7. Voila: Both general and special relativity verified at the same time!?

But Ron Hatch, chief designer of the GPS, seriously questions this statement.  I do the same in Many-Minds Relativity.

Satellite clocks are continuously synchronized to a refernce ground clock and thus any small initital adjustment of clock frequency would get compensated by the synchronization. In particular, even without initial adjustment, GPS would work fine, as far as I can understand at least.

The fact that a certain combination of special and general relativity suggest exactly the 38 nanosecond offset claimed to be what is required to make GPS work, seems a bit too good to be true, in particular since all 24 GPS satellites follow different trajectories. It seems more plausible to believe that any small offset would be compensated by continuous synchronization so that just about anything could be proved by GPS. Or what do you think?