For a closed and sealed cooler, the source of heat that is melting the ice is the cooler walls.
The material property that favors draining is thermal conductivity.
Air behaves as an insulator from the heat source (cooler walls) with k= 0.02 W/mK whereas water is more of a conductor with k= 0.6W/mK
So the melt water in contact with the ice blocks and some area of the cooler walls will transfer heat to the ice blocks 30 times faster than air.
Draining will replace that conductive water with insulating air and the ice will melt at a lower rate.
You can demonstrate this by putting an equal amount of ice in a couple plastic bags and hang them in the air.
Punch some holes in the bottom of one so the melt water drips away and leave the other holding the melt water.
You'll find the draining bag will have solid ice after the other bag is full of just liquid water.
Personally, I just freeze those 2.5gal drinking water containers in my chest freezer to -10F.
The day before leaving I put one in the empty cooler (128qt Galaxy) to pre-chill.
Then the morning we're leaving pack it with two of the frozen jugs (5gal ~ 40lbs. ice) and keep as much of the food / drink going into the cooler frozen as is practical.
With good cooler management (open only once or twice per day, ideally in the morning) I've had it keep food cold for over a week with no issues.