why would wrapping ice and putting it into the icebox inhibit melting?

Answer 1

Both wrapping thge ice in paper or plastic foam, and putting it into an icebox insulates the ice from heat from the outside. Iceboxes of old times have plastic foam layers that are poor insulators of heat; They keep heat from getting in. It takes 80 calories per gram of ice to melt the gram of ice from ice at 0degC or 32degF to water at the same temperature. Whatever you can do to keep those 80 calories out will keep that gram of ice.

Answer 2

In the icebox (refrigerator?) ice doesn't melt; it sublimates from solid to vapor. The wrapping helps slow this process.