>relativity must be taken into account for GPS to work. If relativity wasn't a thing, GPS would still work

That statement is logically inconsistent.

The paper I cited above

https://www.astronomy.ohio-state.edu/pogge.1/Ast162/Unit5/gps.html

describes the role of relativistic "time dilation" in the functioning of the GPS coordinate system

https://en.wikipedia.org/wiki/Time_dilation

relative speed and the relative strength of a gravitational field each affect the local measurement of "time."

In the case of a GPS satellite, which is out in space and farther from us (its users -- and the earth as a gravitational well) and moving faster relative to us (because they need to stay in orbit and constantly fall over the horizon while we are stationary on the ground), these relativistic effects work at cross-purposes.

GPS uses not triangulation to determine a location, but tri-lateration with a fourth satellite to account for timing delays due to relativity.

The paper I cited notes " If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day"