The 'chunky convex' is more about a very thick primary grind behind the edge, which is what will degrade cutting performance. That thick steel behind the edge also makes it more difficult to sharpen at all, whether convex or v-edge, because the thick steel behind the edge precludes any chance at acute sharpness - the edge angle will always be too wide with such thick steel behind it. That is what will result in dull edges, or edges that don't cut well even after they're sharpened.
For very thick convex blades, the better strategy, which really only needs doing once, is to thin the primary grind out. That's where laying the blade lower to the sandpaper to thin out the grind behind the edge will make the bigger difference. BTW, that's even better-done with a hard backing under the sandpaper, like wood or stone or glass - it'll still convex somewhat when done freehand, but to a thinner and more subtle convex, which will improve cutting performance. Then sharpen the edge as you would any other knife, on a stone. Thinner grinds are much, much easier to sharpen and much easier to maintain sharp, no matter whether the grind is convexed or not.
And a point to remember is, if you're doing the sharpening by a freehand method, whether with a stone or sandpaper, the edge geometry will still convex somewhat, due to the inevitable and natural variability in the held angle. The odds of the edge being sharper will still be better when when it's done on the hard stone, which is what will prevent too much edge-rounding as can happen too easily on softer sharpening substrates.