A product designed specifically for cleaning phenolic balls is Aramith Billiard Ball Cleaner, developed by the same people who make Aramith and Centennial balls (the Saluc company in Belgium). It costs approximately $10 for a bottle of about 8 ounces.
To use the product, just put some on a cotton cloth or a microfiber cloth, rub it on the balls, let it dry a bit, then wipe off the haze with a clean, dry cloth. The balls come out looking great. An entire set can be done in just a few minutes, and it makes a ball-polishing machine for home usage completely unnecessary.
Under the WPA equipment rules, it is actually illegal to use a wax or a "polishing or rubbing compound" on the balls. Here is the language: "Balls should be unpolished, and should also not be waxed. Balls should be cleaned with a towel or cloth free of dirt and dust, and may also be washed with soap and water. Balls contaminated with any slippery substance - treated with a polishing or rubbing compound and/or waxed - must be cleansed and dewaxed with a clean cloth moistened with diluted alcohol before play." It is my understanding that the Aramith Billiard Ball Cleaner contains no wax, but I wonder whether it passes muster regarding the other terms in the rule. It would be strange if it didn't!