Perhaps the more modern view on physical theories is more convincing for you than Newton's quasi-axiomatic approach? As I said before, Newton's postulates are deduced from observation. Famously Newton said "hypotheses non fingo" (I don't invent hypotheses).
The modern point of view is to use the action principle to formulate fundamental dynamical laws and employ symmetry principles to constrain the possible action functional. For Newtonian mechanics the symmetry is Galileo symmetry of Newtonian spacetime. It postulates that space and time are homogeneous (i.e., no point in space is preferred compared to any other and no point in time is preferred compared to any other), space is Eulidean and thus also isotropic. Finally there exists an inertial frame, and it's not possible to distinguish any inertial frame from any other, i.e., the physics is also invariant under Lorentz boosts (i.e., changing from one inertial frame to another one moving with constant velocity relative to the former). This implies 10 conservation laws according to Noether's theorems, and the generators are energy (Hamiltonian) (time translation invariance), momentum (space translation invariance), angular momentum (isotropy of space), and the center-of-mass velocity (boosts). In this approach the somewhat problematic idea of "force" is a derived concept, and the form of the laws is founded on symmetry principles.
This idea is crucial for an understanding of modern theories, particularly special and general relativity which introduce new space-time models and quantum theory, where the symmetry principles define the operator algebras of observables.