In that answer on Quora, you conclude:
> Seen this way defining $L=T−V$ looks like a bit of a hack to tidy up the equations for a conservative system rather than something fundamental. One can just use $L=T$ at least for classical mechanics.
Yet nature treats it as fundamental, because this is the magnitude whose path integral gets minimized.
Perhaps kinetic-minus-potential-energy is a primitive magnitude for which our brains have not evolved an intuition -- a foundation for a more abstract view of nature. So it would be fundamental, but easy to relate to the primitive concepts that biology has pre-wired us to understand, like pushes, pulls, and speeds.
If someone were to tell me that in a system of colliding particles, the sum of the masses times velocities is always preserved, I might at first wonder whether there is some meaning to the product of mass and velocity -- or if this was just one of nature's abstract invariants. But we've given a nice name to that product, and it can be related to a Newtonian intuition about how much "oomph" a moving mass has. Furthermore, it is intuitive that oomph is proportional to mass, and also to velocity. And, it even "makes sense" that the total oomph in the system is preserved.
But $m v$ could be on an equal footing with $T - V$, in terms of how intrinsically fundamental they are.