The question about minimizing potential energy and the replies that such questions do not make much sense is a typical conversation between a physicist and a mathematician:
Physicist: - Why systems tend to minimize potential energy?
Mathematician: - Look around, lots of things follow this principle: potential energy, entropy...
Physicist: - OK, I can see that, and that's why I'm asking. Why? Why lower number is better? What's so special about that? Why not bigger numbers?
Mathematician: - Don't ask such questions. This was proved by experiments. We just must accept it as a fact, as an axiom. And axioms just are. So don't get metaphysical.
... and this is where the poor inquisitive physicist quits asking any further if he wants to pursue his career, as the label of a "philosopher" might not be very helpful ...
Well ... the mystery of the tendency to minimize potential energy evaporates quickly as soon as you ask yourself other questions: What this potential energy comes from? What it represents? What is the underlying source/cause?
In the case of the ball rolling down the slope the answer is obvious. This cause is gravity. Massive bodies simply attract all other massive bodies around, and since gravitational potential is defined in terms of distance to the source (centre) of gravitation, then obviously the ball that is moving toward the centre of gravity is minimizing its potential. So the important question here is: what is more fundamental - gravity or gravitational potential? Is the axiom of minimizing potential some ultimate answer? Or ultimate cause? In case of gravity, the potential just tells you in a way how distant the attracting bodies are, and therefore whether the force can produce any movement (and how much of this movement) or not. Minimum potential means that no movement due to the force is possible; maximum potential means that (a lot of) movement is possible. And the source of movement is the force, not the potential. And the movement caused by the force can only go from larger numbers (distance) to smaller numbers. That's how it works: the force pulls things in, so the distance becomes smaller. And that's how the potential has actually been defined. Therefore the answer is not that Nature (for some metaphysical reasons) developed a liking for small numbers. It is rather that Nature provides this very fundamental attractive force - called gravitation - which makes matter "come together". And the tendency to lower the potential is a way of saying that the attracting force causes bodies to come together, and thus lower the distance between them.
To sum up our gravity case: The tendency of a system to minimize potential energy results from underlying force pulling a body toward its source. (or pushing a body away, if the nature of the force is repulsive). This results in minimizing distance, therefore in minimizing potential energy. So more potential energy equals more potential movement, and less potential energy equals less potential movement. (Obviously, if the two attracting bodies are already together, the force is not able to produce any movement, hence zero potential energy.)
Now, this reasoning is equally true for electric potential energy or elastic potential energy. In all of these cases there is an underlying force that makes matter move. So if the nature of the force is attractive, the related potential energy grows with distance, which represents potential movement these forces can produce.
So you were perfectly right to ask "why" about this axiom. This way you are more likely to produce more profound answers that will lead you to more fundamental phenomena, and ultimately to better understanding Nature. There is probably a limit to understanding why things are the way they are, but the tendency to minimize potential energy is certainly not the Ultimate Cause.
By the way, you said you would rather not get the answer in terms of entropy. You were right, because entropy is not really necessary to find the answer about the potential. Yet the case with the ball shows us a very interesting thing - gravity and entropy are two basic opposing phenomena. Gravity counteracts entropy. Gravity pulls molecules together, while entropy makes them move away. It's easier to understand this after the term "entropy" is clarified and demystified. Entropy is simply the phenomenon associated with the movement of molecules (or particles). Molecules with higher energy, i.e. moving faster tend to travel to regions where there are more molecules with lower energy, i.e. moving slower. Why? Sure you can ask why, this is also no ultimate axiom or metaphysics. This is simply because molecules moving faster are more likely to bump each other when they are grouped together. But they are less likely to bump those moving slower. The consequence is that fast moving molecules are restricting each others movements, as they hit each other more often. Therefore, the fast moving molecules will more often penetrate areas where there are a lot of slower moving particles. In any given time period the slower ones "occupy" less space thus leaving more room for the fast ones to move (in a given $\Delta t$ necessary for a fast molecule to travel $\Delta x$ that equals its size, a slower particle will travel less distance than a faster one). And when the faster molecules mix with the slower ones, they will give some of their speed (energy) to those moving slower (through collisions, which also do occur although less often). This way the system goes from "higher entropy" to "lower entropy". But this happens not because Nature just likes small numbers better, as you can see. It can be perfectly explained mechanically without assuming axioms.