The principle of the minimum action of Maupertuis was true only in some restricted cases, and it was false in most mechanics problems.

Hamilton has introduced a new physical quantity, for which he has used only the name "function S". He formulated a variational principle for the "function S", analogous to the principle of the minimum action, but from which it is possible to deduce the system of equations of Lagrange, so it has general applicability.

Hamilton's "function S" is relativistically invariant and nowadays it is usually called Hamilton's action. It is the integral of the Lagrangian, not of the kinetic energy, like the traditional action. It is proportional with the phase of the wave function in quantum mechanics. In relativistic mechanics, the Lagrangian is the component of the momentum-energy that is tangent to the trajectory in space-time, so "function S" is the line integral of the momentum-energy over the trajectory in space-time.

So Hamilton's variational principle is quite different in meaning and applicability from the principle of the minimum action that existed before him. It remains true in all forms of physics that have been discovered after Hamilton.

Why is the general intuitive understanding of these things so rare that it is not even the norm to teach it?

(I have yet to encounter an explanation of the Legendre transform that convinces me the person actually understands it as something more than runic manipulation)