I reached the same conclusion, Coach; I use my sculling paddle after I get close to where I want to fish. I almost always fish tenkara these days, so can easily cast and scull at the same time.
Unlike you, I do not break down my kayak paddle, just lay it athwart the kayak above my lap. There is no loose line to tangle with the paddle because of tenkara. My little kayak is only 8 ft. long, and stowing the broken-down paddle would pose a problem.