You're making a big assumption that there is a significant heat source transferring heat to the water other than through the air. Otherwise in both cases you still rely on heat transfer from the air, it's just that in one case it has to heat up the water first. In a typical experimental setup this might be from a bowl sat on a countertop at ambient temperature or something, but if the bowl had a very low thermal mass and was supported by very thin members then the draining setup could easily melt the ice quicker.