Ok, so why do atomic clocks on satellites in orbit tick more slowly than the clocks on the surface of the Earth? what you're saying is that from the satellites frame, the clock on Earth would tick more slowly as well, and they would both cancel out, but that's not what we observe or measure?
GPS satellites have an added complication. Their speed makes time go slightly slower, but they are further away from the Earth's gravity well which makes time run slightly faster.
But we do observe exactly what Einstein predicts, otherwise GPS wouldn't work at all.
Watch the video I posted, it explains how two observers can BOTH see the other as being slower, each measuring the other and both coming up with different results, both being correct from THEIR frame of reference. Time does not work the way one expects once you start messing around with relativity. In fact, it's entirely possible for two observers to observe far away events and see them happen in a different order. Who is right? Again, both... different frames of reference (speed or gravity) will produce different observations.
It's very weird, but has been proven over and over again in thousands of experiments. Relativity is one of the most tested theories out there, and has never failed to work exactly as predicted. So far. But that's science, when someone finally does break it, we will just modify it and have an even better and more accurate theory.