But it doesn’t, which is the reason why it happens. Because that intuition of “oh, there are two phases, this means order and entropy is disorder so the entropy is lower” doesn’t correspond to the reality of the physical concept of entropy (the thing that is maximized for a thermodynamic system In equilibrium). That’s the point of the article.
Note that the discussion about potential energy is a red herring, phase separation does also happen in absence of gravity (you get droplets which don’t coalesce as easily, but it happens nevertheless).
And the reason it happens is because it doesn't magically get the corresponding external energy. However if we artificially, like gods, arranged everything to be non layered but the same temperature, the entropy would be higher than if layered and same temperature.
I mean that ceteris paribus the layering (separation of materials) causes a decrease in entropy. However, the system still does this, because it gains more entropy through a different "channel".
If we could destroy the layering, thereby evenly mixing things up, but not changing anything else that affects entropy, we could increase the entropy. So it show that the layered version is lower entropy as long as all else is the same. The catch is, this would require adding extra energy into the system, because the mixed version has more potential energy. We could also imagine a mixed up configuration that has the same total energy, but then the temperature would be lower since more of the energy would be used up as potential energy, leaving less for heat. In that case the entropy would be lower.
Main point: the layering, just the fact that it is layered, this property, if we know nothing else, is still something that is a low entropy thing. So our intuition is not broken that order corresponds to low entropy.