If you are interested in doing classical physics, then Newtonian mechanics and Lagrangian mechanics are equivalent formulations. (At least when their domains of applicability overlap -- there are equations of motion you can write down which don't follow from a Lagrangian so in that sense Newtonian mechanics allows for more general equations, and that can matter in practical applications, but as far as we know to date, at a fundamental level, physics can be described with a Lagrangian). Historically it may have been difficult for humans to discovery Lagrangian mechanics without knowing Newtonian mechanics, but logically you can start from either formulation and show the other is equivalent.
If you are interested in philosophical interpretations of classical physics, then indeed Lagrangian mechanics and Newtonian mechanics might suggest different philosophical viewpoints. However, this is not a physics question, in that this kind of interpretation of the formalism cannot be decided by experiment. In terms of observable quantities, both formulations give the same predictions.
If you are interested in generalizing beyond classical physics, then Lagrangian mechanics naturally leads you into the path integral formulation of quantum mechanics. There's not really a clean, direct way to go from Newtonian mechanics to quantum mechanics. This illustrates a point made by Feynman (and probably others), that having equivalent formulations of a theory is useful when you try to go beyond that theory, because one of the formulations might generalize to the next layer of abstraction more easily than another.