I think people usually use oilseed radish or tillage radish for a cover crop. both do slightly different things and while they can both be used the same way one goes deeper with thinner
roots and one I think the roots are fatter and wider. both sold in bulk for use as green manure or for cover crops cost less so "cheaper" than the daikon radish that has been bred for flavor to eat although those radishes will work as well they just cost more per seed since they are used for human food so generally less is grown. this is my understanding of using 'daikon' radish for soil improvement. I don't know about when to slash and it sounds like Karl does!