You cannot repair a bead- or sand-blasted finish without expensive equipment and media that match the original factory choices. You cannot spot-repair such damage, either. The whole part has to be redone.
Blasting means the metal is bombarded with tiny balls of material, that leave a matte grey finish. No lines. Some folks call that satin, but it's a non-reflective, non-directional finish. The surface is basically pitted, on a microscopic level, so light is reflected in a diffuse manner. It also means the metal is more likely to corrode or stain, because of the increased surface area of the pitted surface.
Satin or brushed finishes are exactly that -- a fine abrasive is used to scratch the surface of the metal, in ONE direction, leaving a visible line pattern, and a less reflective surface than a bright-polished finish. You can repair such finishes using matching abrasive brushes or "sandpaper." Bergeon also sells a glass-fiber scratch brush, which looks like the item in your links. But using anything like that takes great care and practice, or you'll make it worse. Most proper refinishing means taking the item apart. The brushes are really hard to use, and can easily make things far worse.
Bright-finished "polished" surfaces have lines, too, but they're really microscopic and not visible to the naked eye.
Polishes don't really work well on metal that has been brush or satin finished, because the grit level of the polish almost never is right, and the Cape Cod stuff is polish in cloth that is hard to control for one direction. If you want to restore a bright polished finish (no lines), you can and should use various levels of grit in polish. Sometimes you can match the original grain by trying out various levels of abrasives, such as 1000-1500-2000 grit you see with metal-working papers. I've regularly used MicroSurface abrasive fabrics to repair scratches on the visible part of deployment clasps, but that's a small part, and you have to do the whole part over at once -- and it's not a watch case with multi-faceted machining! You can do this on bracelets, too, if you're very careful to mask parts and have hours to spend.
If you polish a satin surface (or any surface), you'll be removing some of the surface, which is what any polish does. You may be adding some new scratches in the direction you want, but you will not end up with the original finish, unless the original finish was done in the same way as the repair.
On close inspection, the Cape Cod stuff doesn't bear up to a finicky standard, IMHO.
