Well, I'd say the Hamilton principle of least action (I'd rather call it stationary action), is just a method to formulate the dynamics in a broad range of physical systems. On a fundamental, to our present understanding, all of physics can be formulated in this way.
Further it turns out that the Hamiltonian formulation, i.e., dynamics in the (locally) symplectic phase space is the true mathematical framework in fundamental physics, and it's the most convenient starting point for the heuristics of quantum theory ("canonical quantization").
In other words, the Hamilton principle is a kind of ordering scheme of the possible dynamical equations to describe physical systems. Another important point are that it enables an elegant and natural scheme to describe symmetries. Since Einstein's famous first sentence in the famous relativity article of 1905, the analysis of symmetries of the natural laws in terms of Lie groups and Lie algebras (and, more recently, their generalization to ##\mathbb{Z}_2## graded algebras, aka "supersymmetry") has become of prime importance. Among the very rich possibilities to invent equations of motion derived from the variational principle those have been found most successful in describing the physical world which are constrained by symmetry principles.
All of theoretical physics, if presented a posteriori in a deductive way in a "quasi-axiomatic" formulation, starts with a (so far classical) spacetime model which implies some geometric structure and thus also symmetries. So for any physical theory, using a given spacetime model, they should obey the symmetries of this spacetime model in order to be consistent with it. Since Lie symmetries imply conservation laws (one of Noether's theorem) each one-parameter subgroup of the spacetime symmetry group implies a conservation law, and in Newtonian or special-relativistic space time this implies the 10 conservation laws (energy, momentum, angular momentum, and center-of-mass/momentum motion), valid for any closed system (in General Relativity, where Poincare symmetry is realized only locally, this holds only locally).
Last but not least, when making the heuristic step from classical to quantum theory, the symmetry principles are the only systematic way to (admittedly a posteriori) understand, why the quantum theories look as they do, since the operator algebras governing the specific structure of either non-relativistic QM or relativistic QFT. Canonical quantization has been the way to heuristically find the concrete theory of QM and relativistic theory, but the symmetry principles tell us, why these should be the inevitable consequences of "quantizing" a classical theory. In some sense they are the true "correspondence principles" making the heuristic way of "quantization" possible, and canonical quantization is also not a safe ground to study more complicated systems. Already the theory of quantizing the rigid body (solid top), leads to wrong equations when done naively without the guide from the rotation group, finally leading to applications in, e.g., analyzing the rotation spectra of molecules, etc.
As it turned out, also other even more abstract symmetry principles are of great importance in all physical theories, most importantly in elementary-particle physics, where the analysis of the empirically found conservation laws, concerning "intrinsic quantities" like electric charge, baryon and lepton number, etc. plays the role of an ordering principle, bringing order in the plethora of phenomena, particularly the "zoo of hadrons" found in various particle accelerators.
Of course, Hamilton's principle alone is a pretty general but empty scheme to formulate and then analyze dynamical laws, formulated in terms of differential equations, leading naturally to symmetry principles (thanks to Noether's work of 1918). Physics after all is an empirical science, and you need a lot of observations to figure out the right symmetry principles to finally formulate a theory like the Standard Model of elementary particle physics that can be taught within 1-2 semesters in our physics curricula, summarizing decade-long empirical and theoretical work!
