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In this answer I first give a link to a post by me on physics.stackexchange in which the question 'why should the time integral of T-V be stationary?' is addressed. The rest of the answer is focused on philosophy of physics.
Hamilton's stationary action demonstrated with visualizations. The diagrams visualize a trajectory, variation sweep of that trajectory, and how kinetic energy and potential energy respond to the variation sweep.
Does the PLA tell us something about the Universe?
Let me make a comparison:
As far as we can tell the entire domain of physics has the following in common: the physics taking place can be described with differential equations. There is something universal about differential equations.
It seems to me that differential equations are ubiquitous in physics because a recurring theme is that some physical quantity A relates to the rate of change of some quantity B. In theory of motion there is the two-step cascade of position, velocity, acceleration; a cascade of first derivative with respect to time and second derivative with respect to time.
In and of itself that universal aspect is not enough to constitute a Principle of Physics. There is no such thing as offering 'the Principle of Differential Equations' as 'the closest thing we have to a theory of everything.'
We have that every phenomenon that is modeled with a differential equation is modeled with a bespoke differential equation.
In parallel with that: whenever a stationary action approach is applied the quantities that go into the equations and the relations between those quantities are specific to the phenomenon that the approach seeks to solve for.
Differential equations are a mathematical tool, and likewise a stationary action equation is a mathematical tool.
At this point I want to emphasize the following: the name 'least action' is awkward because it represents an over-interpretation. The purpose of the stationary action equation is to identify at what point in variation space the derivative of the action is zero. Whether that point-where-the-derivative-is-zero is a minimum or a maximum is irrelevant. (There may be cases where it can only be a minimum, or can only be a maximum; that is on a case-by-case basis. It's just never relevant whether it is min./max.)
The name 'stationary action' expresses the idea that when framing a theory it is better to not let the assumptions go beyond necessity. As a matter of principle the name 'stationary action' says only: look for the point in variation space where the derivative is zero.
In classical mechanics there is the concept of Energy. A force is categorized as a 'conservative force' when it has a well defined corresponding potential energy. (The potential is well defined if the difference in potential between points A and point B is independent of how an object moves from A to B)
With a force that is conservative: acceleration due to such a force has the property that the sum of potential energy and kinetic energy is conserved.
That is where the stationary of stationary action comes in: during the entire time the rate of change of kinetic energy must match the rate of change of potential energy.
When those two rates of change match during the entire trajectory then the derivative of the corresponding action is zero.
A recurring theme in physics is that of conserved quantities.
In classical mechanics: the point in variation space where the derivative of the action is zero identifies the point in variation space (the true trajectory) which has the property that everywhere along the trajectory the rate of change of kinetic energy matches the rate of change of potential energy.