You're simply wrong. There's no other way to put it. The GPS system would have been simply impossible to deploy without the general theory of relativity. There's no extreme energy or mass involved, just precision requirements that are influenced by the minuscule differences in time experienced by the surface of the earth and orbiting satellites.
Also Newton's laws famously could not account for Mercury's orbit. Mercury is just an ordinary planet orbiting an ordinary star. Nothing extreme is involved. He knew his laws were incomplete. But they were so dead-on in basically every other scenario that could be physically observed at the time that he figured there was some small tweak missing (or maybe another planetary body that hadn't been spotted yet).
Also Newton's laws famously could not account for Mercury's orbit. Mercury is just an ordinary planet orbiting an ordinary star. Nothing extreme is involved. He knew his laws were incomplete. But they were so dead-on in basically every other scenario that could be physically observed at the time that he figured there was some small tweak missing (or maybe another planetary body that hadn't been spotted yet).