The lag time is central to the operation of GPS. Time is distance. The distance puts you on a sphere centered at the satellite's location. Make 3 such measurements and you can get a single intersection of the 3 spheres, which tells you where you are. However, in order to measure those distances, you must measure that time very accurately, and that takes a 4th satellite. More is better, because none of the measurements is perfect, and because of something called "Geometric dilution of precision", which deals with "Good" and "bad" geometries of the satellites used to obtain the solution. In general, it tends to affect the measurement of altitude, but it can also affect position on the map.
Right, thus the “which must also be taken into account” comment. But such time lag is a separate matter from the relativistic effects (i.e. everything you mention would still be true even if we weren’t taking relativistic effects into account.).
I wasn’t sure if your post was made simply to add to what I said or if you thought you were disagreeing with me. :-)