If we have a chain of fixed length hanging from two points we know that it will form a curve that minimizes the chain's potential energy.
If we imagine the chain as having many small segments, then the potential energy of each segment is $E_p = mgh$. As the number of small segments approaches infinity, their masses equalize because the difference in mass between any two segments goes to $0$ as the number of segments goes to infinity.
So if we want to minimize the total potential energy of the chain, we need to minimize the sum of potential energies of every segment as the number of segments goes to infinity. However, since we know that in the limiting case all small segments have the same mass, minimizing the sum of their potential energies is equivalent to minimizing the sum of their heights (as the number of segments goes to infinity).
This limit of a sum is by definition the integral of the chain curve. Therefore, to minimize the potential energy of a hanging chain, the area under the chain must be minimal.
However, this is not the case because the curve that minimizes that area is known to be a semicircle. This would mean that the shape of a hanging chain is a semicircle which is obviously false.
What is wrong with my argument? I don't see a reason why minimizing the area doesn't minimize the potential energy. By the way, I don't know physics above high school level, so I would be very thankful if someone can answer this without super heavy math. Calculus is fine.