So the principle that systems find themselves at an energy minimum needs to be heavily qualified: it is a consequence of ever-present friction/drag forces having a sign opposite to your velocity, so that the power wasted due to friction is terminally negative.
The principle therefore states, "when a system only has two forces, one due to drag and the other due to a potential energy, the system tends to end up at a minimum of the potential energy."
There are many situations where it does not readily apply. One of the most obvious is air molecules in our atmosphere! The prediction of the principle is that the atmosphere should all fall down to the ground: obviously that's not happening! The basic reason why involves a more detailed understanding of thermodynamics: the friction in this case transfers energy to the molecule's surroundings, which is either the ground or other air molecules. In turn, "kicks" from the thermal motion of the ground and other air molecules keep those air molecules suspended in midair.
Another situation where it doesn't apply is a rocket blasting off. Here, there is another force: the force of the hot gases that the rocket is expelling out its rear end. As long as those forces continue, that rocket's not going to find its way to the ground. If those forces stop, then the rocket may nevertheless have enough kinetic energy to leave the solar system. (One could fix the principle by saying that it either ends up at a potential energy minimum or else "at infinity", perhaps.)
Another situation where it doesn't apply is static friction. There are lots of things on my desk that, if humanity disappeared tomorrow, would I suppose eventually get jostled to the floor as plants and animals reclaim Kansas City. But my desk stays together because of static friction of screws, and the stuff on my desk stays on my desk due to its static friction with that surface. Static friction holds things together in such a way that there is no motion, hence there is no drag force, hence the power is not being dissipated.
As you have noticed, there is a "combination" of these other systems, burning food chemically to engage muscles so that a biological system can use static friction to ascend a mountain, where we end up violating the principle. That's OK, because it was just a useful little rule of thumb in the first place. It doesn't have the status of a fundamental Theory of Physics or anything.
There is a principle which is a little like it, known as the Second Law of Thermodynamics, which does indeed apply here too. One way to phrase it is, "to transfer the energy in a degree of freedom further away from the average, other degrees of freedom always need to go closer to the average commensurately." Our uncertainty about where energy is in the various degrees of freedom of a system can be quantified as a number called entropy. The way that you increase your gravitational-potential-energy degree of freedom when you climb up a mountain is by taking stockpiles of chemical energy in the form of food and stored fat, burning them up (lowering their energy to be much closer to the environment's) with a biochemical process which causes your muscles to store and expel energy, and you use your intelligence and a concerted motion of these muscles to push yourself up the mountain (using those muscles' energy to raise the gravitational energy of your center of mass). Indeed, you cannot do it if you are severely malnourished.
