In a way, the action is really best understood in a quantum-theoretical context, where it naturally emerges from Feynman's principle (essentially, one evaluates matrix elements of the time evolution operator $U(t',t)={\rm e}^{-i\hat{H}(t'-t)}$ with $\hat{H}=T(\hat{p})+V(\hat{q})$ by applying the Lie-Trotter formula, inserting complete sets of eigenstates of $\hat{q}$ and $\hat{p}$, and integrating over the eigenvalues of $\hat{p}$, which leaves one with a time-discretized path integral containing (a discretization of) $\int L(q,\dot{q})\,{\rm d} t$, and in the limit one obtains the usual Feynman path integral).

There is a natural connection from this to the principle of least action by considering quantization as the inverse in some sense of tropicalization: where quantum theory assigns each path an amplitude $U\in\mathbb{C}$ such that concatenation of paths corresponds to multiplication of amplitudes and multiple paths interfere via addition, the classical theory assigns each path an action $S\propto\log U\in\mathbb{R}$ such that concatenation of paths corresponds to addition and multiple paths interfere via taking the minimum. There is a body of work along those lines by John Baez and others.

In a way, the action is really best understood in a quantum-theoretical context, where it naturally emerges from Feynman's principle: essentially, one evaluates matrix elements of the time evolution operator $U(t',t)={\rm e}^{-i\hat{H}(t'-t)}$ with $\hat{H}=T(\hat{p})+V(\hat{q})$ by applying the Lie-Trotter formula, inserting complete sets of eigenstates of $\hat{q}$ and $\hat{p}$, and integrating over the eigenvalues of $\hat{p}$, which leaves one with a time-discretized path integral containing (a discretization of) $\int L(q,\dot{q})\,{\rm d} t$, and in the limit one obtains the usual Feynman path integral involving the action.

There is a natural connection from this to the principle of least action by considering quantization as the inverse (in some sense) of tropicalization: Consider $M$ as a category with points as objects and paths as morphisms, and consider dynamics as coming from a functor from $M$ into a rig; where quantum theory assigns each path an amplitude $U\in\mathbb{C}$ such that concatenation of paths corresponds to multiplication of amplitudes and multiple paths between the same points interfere via addition, the classical theory assigns each path an action $S\propto\log U\in\mathbb{R}$ such that concatenation of paths corresponds to addition and multiple paths between the same points interfere via taking the minimum. There is a body of work along those lines by John Baez (cf. these lecture notes) and others.