I think this makes all kinds of sense. Acids like vinegar dissolve calcium carbonate quite rapidly, and this will be doing it with every wash. The first rinse will heat this up and sling it everywhere vigorously for more than enough time. The subsequent steps will get rid of it.
The only problem I see is if the dishwasher carries over too much liquid from step to step. Dishwashing detergents generally make the wash water alkaline. If there is much acetic acid left from the previous step, this cleaning attribute would get neutralized.