I went through many steps to achieve that result. Because of the deep gouges, and the badge recess that I wanted to eliminate, I started with 120 grit in an air tool. Did some work with rollock disks and some with flap disks. 120 grit leaves some pretty bad scratches, of course. Then I wet sanded with 220, 320, 600, 800. Each time, not switching to the next grit until all traces of scratches from the previous grit were gone.
Then I finished them off with a buffing wheel on a 6" buffer using an emery bar. Then I switched wheels and did rouge.
The Harbor Freight buffer was not expensive. Without it, I could have never gotten this result. Buffing aluminum to a show polish requires high speed, (at least 3000 rpm,) and enough power to build some heat in the piece. The polishing bars are abrasives in tallow. It takes some heat to melt the tallow and make the abrasive really work. Trying to use a buffing wheel in an electric drill is not very effective because it just doesn't have the speed and power to make it happen.
Do yourself a favor and remove those cases, then you can do a proper job of polishing them with a lot less work.