In fact, the principle of least action in practice means a stationary action rather than the least one. So it may well happen (in Classical Mechanics) that the action is not minimal, but stationary or maximal, whatever. Anyway, the classical action $S (t_1,t_2)$ is a functional of unknown variables $\tilde{x}_i$, not a functional of equation solutions $x_i (t)$.

Now, we can calculate the action as a function of time $t$ with substituting the known solutions in the integral: $S (t_1,t)$. Its numerical value is of no use in CM.

Now let us turn to QM in the Feynman formulation via path integral over all paths between two fixed points $X_1$ and $X_2$. It is known that the maximal contribution to the transition amplitude is brought with the paths close to the classical trajectory (if any). If the QM system is in a coherent state (see applets with wave packet motions à la Schroedinger), the paths far from the classical ones contribute too little to the amplitude (the action is too big and the integrand oscillations cancel each other). But if the system itself is far from a coherent state, say, it is in the ground or a stationary state (a standing wave), then all paths contribute nearly equally, I would say. The actions are big for them. In this sense the system is more quantum than classical.