Actually SBF, the problem with classical physics was not that it could not explain the orbit of Mercury (being a slowly rotating elipse instead of a stationary elipse). That was a relatively minor issue. A far bigger mystery in the 19th century was where the sun got its energy. But it was only the result of the Michaelson-Morley experiment (proving there was no “ether” or stationary point in the universe) that brought an end to classical physics.
As to whether Newtonian physics is “true”, I would say for all practical purposes it still is. It will give you an extremely accurate approximation so long as you are not dealing with relative velocities close to the speed of light, extremely high gravitational fields, or objects (particles) so small that quantum mechanics comes into play.
Anyway, don’t mind me. Please carry on.