Uh, you know that the only reason your GPS works the way it does is because we've had to compensate for the relativistic effects of the satellites' motion, right?
As for all that other nonsense, I don't even know where to start....
That gets stated a lot by pro-relativists, but it's not actually true. Determining your location via GPS is based on the differences between the time signals of various satellites and not in any way based on the difference between a satellite time signal and the device's Earth surface local time, as is commonly incorrectly believed. http://www.physicsmyths.org.uk/gps.htm
GPS computes time differences between satellites all moving at the same speed relative to each other. GPS doesn't care what time it is where you're standing relative to the satellites. Relativity is irrelevant to GPS.
No, you still have to compensate for relativistic differences between the satellites and the receivers, no matter what your conspiracy-theory website says. Given that the writer doesn't even believe that stars have fusion reactions inside them or that gravitational lensing occurs, I don't think it's a credible source. Here's a page
that describes how it actually works, written by someone who has been directly involved in running GPS.