The principle of Least (Stationary) Action (aka Hamilton's Principle) is derived from Newton's axioms plus D'Alembert's principle of virtual displacements.
Because D'Alembert's principle allows to account for the (reactions of the) bonds between the components of a system in a transparent way, the Lagrangian and Hamiltonian formulations are possible.
Note1: Newton's axioms, as given, cannot derive neither the Lagrangian form nor the Hamiltonian as they would need the reactions of the bonds to be added literally inside the formalism, thus resulting in different dimensionality and equations for the same problem where the (reactions of the) constraints would appear as extra unknowns.
Note2: D'Alembert's principle is more general than the Lagrangian or Hamiltonian formalisms, as it can account also for non-holonomic bonds (in a slight generalisation).
UPDATE1:
When the forces are conservative, meaning derived from a potential $V(q_i)$ i.e $Q_i = -\frac{\partial V}{\partial q_i}$, and the potential is not depending on velocities $\dot{q_j}$ i.e $\partial V / \partial \dot{q_j} = 0$ (or the potential $V(q_i, \dot{q_i})$ can depend on velocities in a specific way i.e $Q_i = \frac{d}{dt} \left( \frac{\partial V}{\partial \dot{q_i}} \right) - \frac{\partial V}{\partial q_i}$, refered to as generalised potetial, like in the case of Electromagnetism), then the equations of motion become:
$$\frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q_i}} \right) - \frac{\partial T}{\partial q_i}-Q_i = \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q_i}} \right) - \frac{\partial L}{\partial q_i}$$
where $L=T-V$ is the Lagrangian.
(ref: Theoretical Mechanics, Vol II, J. Hatzidimitriou, in Greek)
UPDATE2:
One can infact formulate D'Alembert's principle as an "action principle" but this "action" is in general very different from the known Hamiltonian/Lagrangian action.
Variational principles of classical mechanics
Variational Principles Cheat Sheet
THE GENERALIZED D' ALEMBERT-LAGRANGE EQUATION
1.2 Prehistory of the Lagrangian Approach
GENERALIZED LAGRANGE–D’ALEMBERT PRINCIPLE
For a further generalisation of d'Alembert-Lagrange-Gauss principle to non-linear (non-ideal) constraints see the work of Udwadia Firdaus (for example New General Principle of Mechanics and Its Application
to General Nonideal Nonholonomic Systems)
I) Actually, it's the other way around. Within the context of Newtonian mechanics, the hierarchy is the following from most to least applicable:
Newton's laws are always applicable.
D'Alembert's principle or Lagrange equations. E.g. sliding friction typically violates D'Alembert's principle.
The stationary action principle $S=\int\! dt~L$, with Lagrangian $L=T-U$, and its Euler-Lagrange equations. E.g. a generalized force might not have a generalized potential $U$.
II) In point 3 we have tacitly assumed that the Lagrangian is of the form $$L~=~T-U,\tag{1}$$
as is customary. $T$ and $U$ in eq. (1) may be viewed as representing the kinematic and the dynamical side of Newton's 2nd law, cf. e.g. this Phys.SE post. The linear structure of eq. (1) also reflects a categorical-like composition rules for how to build physical models out of physical subsystems.
There exist strictly speaking exceptions to the form (1), cf. e.g. this Phys.SE post, but these exceptions often lacks categorical-like composition rules, which make them unsuitable for useful model building.
III) For further details and discussions, see e.g. my related Phys.SE answers here, here, and links therein.
Best Answer
On one hand,
The d'Alembert's principle is formulated in the framework of Newton's laws to start with, which limits its usefulness in other areas of physics.
The d'Alembert's principle is more or less equivalent to Lagrange equations. The latter is arguably easy to apply.
On the other hand,
the stationary action principle is versatile/applies to many different areas of physics.
the stationary action principle does no cope well with e.g. dissipation and/or semi-holonomic constraints. However, this is often not an obstacle in fundamental physics. See also e.g. this Phys.SE post.