How does the GPS time signal achieve an accuracy of <= 40ns relative to UTC? From the official GPS website:

How accurate is GPS for timing?
GPS time transfer is a common method for synchronizing clocks and networks to Coordinated Universal Time (UTC). The government distributes UTC as maintained by the U.S. Naval Observatory (USNO) via the GPS signal in space with a time transfer accuracy relative to UTC(USNO) of ≤40 nanoseconds (billionths of a second), 95% of the time. This performance standard assumes the use of a specialized time transfer receiver at a fixed location.

How is this possible, though, if relativistic time dilation effects (both from SR and GR) are already in the order of 38µs/day which the satellite's time signal clearly has to correct for? (In contrast, such a correction is not needed for mere position fixing, see e.g. this answer.)
I mean, the GR effects assume that we live in a more or less perfect Schwarzschild spacetime (which is clearly not the case) and so the time dilation calculations are merely an approximation. Now I haven't done any calculations but to me it seems very unlikely that the time dilation of 38µs/day is accurate to within 0.1% (= 40ns / 40µs) 95% of the time. (For instance, the gravitational acceleration $g$ at ground level already varies from place to place by 0.5%.)
Now that I'm thinking about it: Could my initial assumption be wrong in that the time in the GPS time signal does not actually come from the atomic clocks on board the satellites but from a ground station (whose signal the satellites merely reflect / amplify)?
But even then I'm still astounded by the accuracy of 40ns, given that there's also a significant delay (far greater than a few nanoseconds) between the time the satellite emits the signal and the time the receiver on the ground gets it. But maybe I'm not understanding the term "time transfer accuracy" correctly?
 A: Your link to the official GPS website has a lot of information about how the system work, including the fact that ground stations can and do check clock accuracy and update satellite clock time as needed. The difference in clock rate built into the system is not the final answer - satellite clock times are tweaked should they get out of spec. So, we can assume that the clocks on the satellites are kept current, including all relativistic effects.
Then one can read through the Wiki article on Error analysis of the GPS system. One thing to remember is that for accurate position fix (which is equivalent to an accurate time fix) the receiver needs to see more than one satellite. That way the path length differences can be (mostly) eliminated - the clocks on each satellite are all the same time and the receiver is told where they are and the rest is math.
On my old consumer GPS, I routinely see position fixes with estimated errors of under 10 meters, or less than 30 nanoseconds. That fits well with the Wiki article above.
