I agree with your general point absolutely. But I was very interested to discover (from Radio Astronomer the Freeper) that Newton is *not* enough for piloting rockets to Mars. He does this for a living. Use Newton and he'll miss his orbit. You've got to use Einsteinian mechanics to travel that distance.
Also GPS devices operate according to Einsteinian physics, or your location would be off by a few metres.
One of the great successes of General Relativity was its accounting for the precession of the perihelion of the orbit of Mercury, a phenomenon that Newton's theory couldn't account for: