What's not the same? The picture for 16:9 DVDs is shrunk / visually compressed into a smaller frame, too, just like its film counterpart. It's not what we're doing, exactly, because we're starting with stuff that's too small and stretching it vertically to fill the space, but this is just a very special case. Usually, working from an HD master, there's more than enough resolution and the visual compression / shrinking is obvious."

This is incorrect. The picture for a 16:9 transfer on DVD is stretched out vertically. A film that is anamorphically transferred is squeezed into a smaller frame by use of optics. In both cases, the aspect ratio is changed. The process that you are referring to maintains the aspect ratio.