I love a good convex grind, on some tools. I think that the convex grind is and was one of the most common grinds. I say this from the many hundreds of people that come through my sharpening workshops. Because of how people move their tool over the abrasive or visa versa we tend not to fix and make perfect flat grinds, we are human and wobble, lots. I am mainly a green woodworker and teach people how to sharpen knives and axes and other green woodworking tools.
The convex grind is useful and I mainly use it on my axes and cooking knives. I tried it out on other tools after Paul first started talking about it but have never really got on with it. I have to agree with the old blokes who wrote about sharpening many centuries ago, simple and effective and large round grinding stones were found on every farm and workshop.
The problem with the convex grind is people have a habit of lifting the tool just that little bit more each time to get the tool sharp. They do not work on the rest of the bevel and so the tool soon becomes usable or blunt. Blunt being a too obtuse grind, the tool may be sharp but the larger bevel angle means the tool gets stuck in the wood or vegetable because of the wedging action.
The best way to sharpen a convex tool is to work first on the sharpening angle, set this and you will have a short flat bevel or bevels, then work on the the rest of the bevel never lifting the tool higher than the sharpening angle.
If you want to sharpen a tool with a convex or hollow grind on a large round stone or just 2 flat bevels I do not care, do what suits you as long as it works.