Draining will be faster.

Let's assuming you have a glass of ice in a room with a constant air temperature.

The air will begin to melt the ice and soon you'll have a glass of ice water. The ice is no longer in contact with the air directly and is now submerged in a water that is only slightly above freezing temperatures. It takes a lot of energy at this point to keep warming, so now thay air has to heat the water first before there is a significant temperature gradient to melt the ice more.

Now, if you have the same amount of ice in a strainer, you have 2 changes going on - when the water is removed, they're now 2 separate systems in this constant air temperature exam, the ice doesn't impact the water and the water doesn't impact the ice. Since the water is being removed from the ice "system", you now have less mass to heat up as well and maintain a higher temperature gradient between the air and temperature of the ice.

While water is a better conductor of heat, heating ice with water that is heated by air is still far less efficient than heating the ice from the air directly.