The V shape makes sense for an ideal chain with all of its mass concentrated at the midpoint (with the rest of the chain being massless).
But now consider a chain with its mass distributed over 3 points, e.g. equal masses at the quarter-way points and the midpoint. The quarter point masses will pull the V out of shape, introducing their own bends into the chain. These new bends must change the position of the midpoint mass, and by symmetry the movement must be vertical. Before we added the new masses the midpoint mass was at the lowest possible position, so adding the quarter point masses must lift the midpoint mass.
This is an easy experiment to perform, using some light thread and a few nuts (the metal ones, not the edible kind :) ).
Now subdivide the chain into 8 sections, add nuts to the midpoints of the new sections, and we'll get new bends. Repeat the process indefinitely, and we get the catenary.
Here are the first few steps of this process, calculated using Python. All of the chains in this diagram have a length of 400 units, the background squares are 10x10 units.
You can see the original SVG version of this diagram on GitHub.