Stropping with compound can improve results, but it's not guaranteed. Some compounds won't perform well, if mis-matched to the steel being sharpened. And it's also possible to 'overkill' the stropping step, with compound that's too aggressive for the steel or the pre-existing condition of the edge. If an edge is already in very good shape, sometimes bare leather is all it needs. You've seen this in your example; you saw significant improvement in your edge with bare leather stropping. A less-refined edge wouldn't have responded as well to bare leather. The more refined an edge becomes, the less abrasives are needed to touch it up. Sometimes, all that's needed is to gently straighten/realign the edge, or to clean up some very fine burrs, either of which can be done with bare leather (or fabric, like denim).
It's always good to try things out though, so you'll develop a feel for what compounds can help fine-tune your edges. Especially after re-sharpening on stones, an edge can almost always benefit from an additional sequence of stropping steps, with gradually less-aggressive compounds (for example, 6, 3 and 1 micron diamond compound, in sequence), and then finishing on bare leather. Different steels will respond in vastly different ways, too. This is why it's beneficial to experiment as much as you can afford to.
