Einstein’s theory did overthrow Newton’s. At least in the sense that it presented a way of understanding gravity that worked better than Newton’s theory.

It had long been known that Newton’s theory, for all its merits, couldn’t be correct because it couldn’t account for the perturbations in Mercury’s orbit. We just didn’t have a better way of understanding gravity until Einstein and spacetime.

Newton’s theory isn’t a special case, it’s an impressively good guess which works well enough on Earth and at non relativistic speeds. But ultimately, it’s still wrong.