By your reading, no theory can ever be known to be well founded, as there is always the possibility of another discovery finding phenomena that aren't explained by the theory. I believe that most people would view science (in the wide sense, not limited only to disciplines that apply the scientific method) as a continuous process of abstraction and refinement. Newtonian mechanics is pretty accurate if you limit yourself to objects of a certain size and a certain speed, a limitation which is reasonable considering those are the sides and speeds we encounter most often in our senses.
Also, yes, Newtonian physics isn't well founded for exactly that reason. Although in its original formulation E&M wasn't described.