Regarding the "why the action is this object" part of the question, I find that the easiest way to think about it is from the Hamiltonian perspective. There you can think of it as minimising energy along a trajectory. From that point, a Lagrangian is just a mathematical trick to express the symplectic structure differently.
But if your question was more about "why minimizing something yields trajectories", I personally would argue this is beyond physics. As an empirical science, physicists have seen this kind of behaviour broadly (optics, classical mechanics, quantum mechanics) and just unified it as an overarching principle.
Finally regarding the proof to newtonian mechanics, I don't have anything handy from the pure Newtonian perspective beyond the usual "minimises the lagrangian and your equations of motions look the same". However, you might be interested in proofs which show newtonian gravity as low energy approximation of general relativity. And since general relativity has a nice action formulation, it all gets nicely tied in.
But if your question was more about "why minimizing something yields trajectories", I personally would argue this is beyond physics. As an empirical science, physicists have seen this kind of behaviour broadly (optics, classical mechanics, quantum mechanics) and just unified it as an overarching principle.
Finally regarding the proof to newtonian mechanics, I don't have anything handy from the pure Newtonian perspective beyond the usual "minimises the lagrangian and your equations of motions look the same". However, you might be interested in proofs which show newtonian gravity as low energy approximation of general relativity. And since general relativity has a nice action formulation, it all gets nicely tied in.
Hope this helps!