I'll put it to you this way by stealing an example straight from stack exchange.

Place a cocktail stick through an ice cube and lay it on top of a glass filled right to the top with (room temperature) water. The submerged half will melt quicker than that on top.

So what if we dial "room temp" down to 33 degrees? Same answer. What if we dial it up to 200 degrees? Same answer.

Air is an AWFUL medium for temperature to transfer. It'll take an hour to cool down a warm beer in the fridge. But it's like maybe 12 minutes submerged in water at the same temp as your fridge.

In a cooler you want the maximum of cold thermal mass but the minimum of heat transfer. THATS why draining a cooler makes the ice melt faster. You're literally allowing the ingress of warm air while actively displacing the cold thermal mass. It's got very little to do with wet vs dry like OP is getting at.