Remember that the melt water is also interacting with the warm air as well. It's not the temperature difference that matters so much as the huge difference in the frequency of interactions between gases and liquids which is why water will outpace air here.

FTR I've done this experiment myself using a consumer cooler and blocks of ice. The portion of the ice that is exposed to the water visibly melts/shrinks faster than the exposed ice. A cube of ice sitting on a sheet of plastic mesh will last much longer than the same sized ice cube sitting on the bottom of the cooler (the little puddle it starts to form almost literally gobbles it up).

Want to cool a can of soda fast? Don't plop it in a pile of Ice alone. Add water to the ice and even though you've definitely added heat to the system the rate of the soda's temperature reduction will be vastly quicker.

Edit: and by cool in my previous statement I don't mean literal measurable change in temperature. If the ice is melting both it and the water are at 0deg C. It is the Heat of Fusion that is being transferred that I'm referring to.