It's also less relevant when solving using the more common method which is time-tagging at the receiver, relativity might be a bigger pain if you are doing time-tagging at the transmitter but since the clocks of different satellites are not synchronized and can differ by as much as 1ms that delta would be more of an issue than relativity.
If you are doing normal GPS accuracy which is 15m you don't need to use relativistic correction in the receiver, is you are solving for almanac data you do not need to do relativistic correction in the receiver, you only really do relativistic corrections when you are using centimeter level accuracy GPS and even then you have considerably bigger corrections to do.
Yes relativity was accounted for when GPS was designed, but it's by no means would send you to China, not even to the wrong town.
What GPS needs to work is for the receivers to be able to calculate a proper geometric range delay or at least be constantly and consistently wrong when calculating the delay for different satellites in the cluster.
Discounting relativity would reduce accuracy, but it would be consistent and constant and so would not present that much of an issue, it's also by far not the biggest offender when it comes to corrections.
The other "relativistic" correction is one done in the receiver and is needed for geometric range delay calculation, basically taking into account that light doesn't travel in a straight path, we've kinda known that since the early days of classical optics - Fermat's principle or the principle of least time.
Now while it's true that we are using the "relativistic" version of this in our calculation solving the classical or even ignoring it all correctly as long as we account for other effects would still allow you to find a starbucks.
Since GPS was initially intended for military use it had other uses such as time keeping that can be used to synchronize communication, encryption and other things these errors can be cumulative whilst navigation errors are often not since the drift would be more or less uniform with all satellites you see.