This is the reasoning behind my statement, which were qualified to classical mechanics.

The Beers derivation shows that

$$\frac{d}{dt} \frac{\partial T}{\partial \dot{q}_i} - \frac{\partial T}{\partial q_i} = F_i = \mathbf{F} \cdot \frac{\partial \mathbf{r}}{\partial q^i}$$

is the equation that is equivalent to Newtons laws.
That already seems to make $T$ a kind of Lagrangian, since this an inhomogeneous Euler-Lagrange equation
This equation holds for dissipative as well as conservative systems.

Conservation of energy is expressed by rquiring $\mathbf{F} = -\nabla V$ yielding
$$\frac{d}{dt} \frac{\partial T}{\partial \dot{q}_i} - \frac{\partial T}{\partial q_i} = - \frac{\partial V}{\partial q_i}$$

Using $L = T - V$ is a device make the conservative form of the equation homogeneous, ie remove the RHS force term.
This is what enables mechanic formulated as a minimum principle.
To me that part of the derivation looks a little contrived though (which of course does not mean a whole lot in the greater scheme of things :) )

Also while it is mathematically very beautiful, I have always found the principle of least action somewhat unsatisfying as a physical axiom
(I am very happy with it as a derived theorem),
and I really like the Synge's geometrical reformulation as
$$\nabla_\mathbf{\hat{v}} \mathbf{\hat{v}} = 0$$,
which is seems like a much smaller leap of faith.
If $T$ is used to construct the metric instead of $L$, Synge's approach yields
$$\nabla_\mathbf{\hat{v}} \mathbf{\hat{v}} = F_i$$
which is still very elegant, and what little it looses in elegance it makes up in physical insight,
namely that deviations from geodesy are due to applied forces.

This is why I tend to view using $L = T - V$ primarily as a technical device to make the math prettier at the expense of physical insight
and to view $T$ based formulations as more fundamental, though this is of course subjective.

This view is specific to classical mechanics, as I qualified in the earlier statements.
My understanding is that the force concept does not really make sense in quantum theories an minimum principles are the only game in town then.
However, my undestanding of quantum theories is much more superficial.