A purely coordinate-free argument might be to think about what happens when you take a curve with a bitangent, and then deform the curve so that the two tangent points come together -- clearly this should produce a flex. We already understand how bitangents give rise to simple nodes on the dual curve. On the side of the dual curve, this deformation would correspond to taking a nodal curve and letting the "nodal part" shrink to a single point. Squint and you get a cusp.
(There used to be also a big computation/handwave in this answer, but on a second reading it was really not illuminating at all.)