Suppose you have a container with a drain at the bottom full of frozen H2O which you leave a room temperature to melt. Will the ice melt faster if you drain the water as it melts?

I would expect that the water, being warmer than the ice, will speed the melting of the ice. However, I would also expect that draining the water would remove allow more air to enter the system and heat the ice faster. Which explanation is correct?

Related question: if the container is a cooler to fill with warm beverages, will the beverages cool faster in a wet or dry ice-filled cooler?