The Newtonian model is wrong in some very important ways. Not just 'not quite right' but 'fundamentally in error.'
The fact that gives good-enough answers for a whole class of problems is misleading. So did the old theory of planetary epicycles.
Relativity and Quantum Field Theory use different models which give accurate predictions for a much wider class of problems. But the models make the difference. The math comes afterwards.
It's true if you make some simplifying assumptions in the equations they reduce to Newtonian mechanics. (Well - relatively does. QFT is weirder. Part of it reduces to plain vanilla Maxwell, but it also includes forces and interactions Maxwell never had to worry about.)
That doesn't mean the Newtonian model is just as good. It means it's been demoted to a simpler toy model you can use for certain problem - as long as you understand it's just a toy.
Science is really about model making, not math or approximation. The math just gives you ways to use a conceptual model for practical predictions.
The concepts behind the model always come first. And the concepts underlying GR and QFT are much richer and more explanatory than the concepts underlying Newtonian mechanics.
Sooner or later there's going to be another update, and relativity and QFT will be demoted in turn. But they'll be demoted by another conceptual revolution expressed in math, not by more math based on the same concepts.