Yes, the aluminium foil does work provided the spots are only spots and the chrome isn’t lifting.
My uncle used fresh newspaper to polish his chrome. Apparently the ink is oily while still fresh. His chrome always looked tops.
Just a note on the aluminium foil’s shiny versus dull side difference. Cooking foil is one of the expensive grades of aluminium because it has to be rolled to an ultrathin layer. However, it can’t be rolled to the thickness as sold on the roll, so it is rolled to double thickness from two sheets then separated after rolling. The dull side is the contact surface between the two sheets. It goes through a rigorous inspection for any pinholes or other blemishes before rolling into the cardboard tube.
The drink cans we readily throw away are made of three different grades of aluminium, but that’s another story.