Plotting directed acyclic graphs while displaying the edge weights and node sizes is a bit of an unsolved problem. I agree that in most cases chord diagrams are more confusing than illuminating, but so are the alternatives.

You see them a lot in biological network / pathway / association studies. I personally hate them, because they're hard to read and comprehend. Many times a heatmap conveys the exact same data in a more digestible way. Alas, chord diagrams can be pretty and make it seems like the analysis is much fancier than it really is.