I assume the wet paper towel speeds up the process by a little bit? I can't imagine it makes that big of a difference and still probably slower than throwing it in the ice tray of your freezer.

it will; it's about thermal contact. the air in a fridge or freezer isn't very good at conducting the heat out of things to the (antiradiator - absorber?) at the top.

The cold towel will absorb heat from both the drink and the room in a room, but only the drink in the freezer. and it will do it much faster than the air would.