You are right that the "maximization of entropy" is not the cause of anything; entropy is an emergent phenomenon, meaning that it is a result of the behavior of an underlying system, in this case some collective. The power of the concept "entropy" is that it is independent of the underlying "substrate" — all collectives follow a path to higher entropy, independent of their specific constituents. The idea to essentially look at collectives from a statistical point of view is so powerful because there is only a minimal set of assumptions about the behavior and interaction of the collective's constituents (if I'm not mistaken, random interaction is the only one). The concept of "entropy" therefore turns out to be a completely universal concept, including the literal sense of the word. (In that it perhaps resembles another abstract concept, that of evolution in a broader sense.)

So if it is never the cause of anything but, on the contrary, always a consequence, why do we sometimes use entropy to explain behavior? Because the rules we found how collectives behave can be used to predict their behavior without having to go through all the gritty details. If we want to predict the eventual temperature distribution in a system or the end state of two gases after we remove a membrane we can simply say "the state will be X because that's the maximum entropy", without simulating the trajectory of $10^{25}$ molecules.

This mental "inversion" can be easily shown in play in classical physics. If we have a ball rolling on an uneven surface we can predict that it will come to rest in a local depression, a place of minimum altitude. Isn't that obvious? It will minimize its potential energy. But of course the ball knows nothing about potential energy or the surface topology beyond the point it is currently on. Both are abstract concepts we use to simplify our mental model of the world. If we are allowed to stay within the realm of Newtonian physics, the ball really only "cares" about the gravitational vector and the surface inclination at each point in time and space and some friction, and its velocity changes according to the resulting forces.

Even though the ball is really very dumb we may say "it wants to be to be at the point of lowest potential energy", even though this is the result of the underlying physics, not the cause.

Similarly, we say "the system moves towards thermodynamic equilibrium", even though this is the result of the underlying physics, not the cause. It is just that it is always the case, and cannot be any other way because the concept expresses a fundamental insight into the behavior of collectives.¹