Can we motivate the principle of least action with the principle of inertia that causes a mass particle to resist changes in its momentum? After all, the principle of inertia is the starting point and the fundamental assumption of classical mechanics, like the Encyclopædia Britannica states.
Leonhard Euler gave the following consideration in his "Additamentum 2" (1744):
Since all natural phenomena obey a certain maximum or minimum law, there is no doubt that some property must be maximized or minimized in the trajectories of particles acted upon by external forces. However, it does not seem easy to determine which property is minimized from metaphysical principles known a priori. Yet if the trajectories can be determined by a direct method, the property being minimized or maximized by these trajectories can be determined, provided that sufficient care is taken. After considering the effects of external forces and the movements they generate, it seems most consistent with experience to assert that the integrated momentum, i.e. the sum of all momenta contained in the particle’s movement, is the minimized quantity.
We also have the following by Pierre Louis Maupertuis in his "Les Loix" (1746):
When a change occurs in Nature, the Quantity of Action necessary for that change is as small as possible. [...] The Quantity of Action is the product of the Mass of Bodies times their velocity and the distance they travel. When a Body is transported from one place to another, the Action is proportional to the Mass of the Body, to its velocity and to the distance over which it is transported.
When a mass particle changes its position by $\Delta s$ then, according to the Galileo's law of inertia, it tends to maintain its momentum $p$ with no change in direction or magnitude, ideally $\Delta p = 0$. That is, when the particle has no net external force acting upon it, a product $p (s_2-s_1)>0$ has the minimum value in the sense that the physically realized straight line is the shortest distance between the successive points $s_1$ and $s_2$. If we consider the influence of an external force, we may assume that the inertial resistance holds for every infinitesimal part of the physically realized overall path. Therefore, the inertia of motion causes a sum $\sum \overline{p} \cdot \Delta s$, where $\Delta s$ is an infinitesimal part of the overall path in 3 dimensions, during an infinitesimal time step $\Delta t$, and $\overline{p}$ is the average momentum on each $\Delta s$, to obtain a minimum value. This motivates the Euler's form of the least action principle, i.e.,
$$\delta \int p \cdot ds=0.$$
Moreover, in the presence of conservative force, the only possible source of momentum dependent kinetic energy $T$ is position dependent potential energy $V$, and vice versa. Now the inertia of motion resists the flow of energy between $T$ and $V$, ideally $\Delta T = 0$. Therefore, the inertia of motion causes a sum $\sum \overline{T} \Delta t$, where $\Delta t$ is an infinitesimal time step during the overall energy flow process and $\overline{T}$ is the average kinetic energy on each $\Delta t$, to obtain a minimum value. This motivates the Maupertuis' form of the least action principle, i.e.,
$$\delta \int T dt=0.$$
The previous results are two sides of the same coin:
$$ \int p \cdot ds= \int p \cdot \dot{s} dt=\int 2Tdt.$$
On the other hand, the previous energy flow between $T$ and $V$ means that $\Delta T = -\Delta V$, and this suggests that we can replace $T$ by $-V$ in the Maupertuis' form. Since in the both cases we seek to find a minimum, we may consider the sum of the two terms ($T-V$ or $2T$ or $-2V$) and this motivates the Lagrange's form of the least action principle, i.e.,
$$\delta \int (T-V) dt=0.$$