My favorite explanation is in [Brian Lee Beers: Geometric Nature of Lagrange's Equations](
It starts with Newton's second law and projects it onto the coordinate basis vectors.
For a single particle this is
$$ \mathbf{F} \cdot \frac{\partial \mathbf{r}}{\partial q^i} = m \mathbf{\ddot{r}} \cdot \frac{\partial \mathbf{r}}{\partial q^i} $$
After converting to components, Lagrange equations just drop out after a little algebraic manipulation.
It is never assumed that $T - V$ is minimized or even special in any way.

The derivation generalizes to arbitrary systems including mutiple particle sysms and rigid bodies,
by replacing the mass term with the inertia tensor in Newtons second law,
In traditional tensor notation with the Einstein convention this is:
$$ F_i = I_{ij} \ddot{q}^j $$
Similar derivation is also in [James Casey: Geometrical derivation of Lagrange’s equations for a system of particles](
Casey wrote a series of follow on papers extending the idea to rigid bodies, fluid dynamics, ...

There is a more advanced version of this idea dating back at least to [Synge: On the geometry of dynamics](, where the inertia tensor is treated as the Riemannian metric on the configuration manifold.
The principle of least action then becomes quivalent to the principle of least distance on the configuration manifold under this metric.
In particular, conservative systems follow geodesic trajectories in the intrinsic geometry of the configuration manifold under the inertia metric.
By itself that is not a huge gain in insight, but in the absence of torsion, geodesic trajectories are characterised locally by $\nabla_\mathbf{\hat{v}} \mathbf{\hat{v}} = 0$ where $\mathbf{\hat{v}}$ is the unit tangent vector of the path, ie geodesic trajectories have a constant intrinsic direction or straightest paths are the shortest.
This is shows the relationship between the Principle of Least Action and Hertz's Principle of Least Curvature.
It also largely takes the teleological voodoo out of the Principle of Least Action.

Geodesics are equivalent to paths of least curvature only in the absence of torsion,
[Gabriel Kron]( and more recently Hagen Kleinert have shown that, in the presence of torsion,
it is the principle of least curvature that is correct.

[Crouch: Geometric structures in systems theory]( also has a nice exposition of geometrical views of dynamics.

This also shows that the relationship of the Lagrangean and Newtonian formulations of classical mechanics is really that of intrinsic and extrinsic approaches to differential geometry.