what actually works to polish silver?

For the best finish, a proper silver polish and a soft cloth, rubbing in straight lines rather than circles. For a big pile of pieces the foil method is quicker: line a bowl with aluminium foil, fill with hot water, add a tablespoon each of baking soda and salt, and put the silver in touching the foil. The tarnish comes off on its own. Skip it for thinly plated stuff or anything with darkened detail you want to keep, because it strips that too.