The rate of heat transfer due to convection depends on a temperature difference between the fluid and the wall of the box that is in contact with the fluid.
One must look at the other important heat transfer mechanism which is the conduction of the heat through the insulation
There are other heat transfer to consider but the most important is the conduction. The rate of heat transfer through the insulation
is a function of the conductivity and thickness of the insulation and the temperature difference between the outside of the insulation and the inside. The inside is roughly at 0 deg C if the ice is slowly melting and there is water.
In the steady-state, lid on box, ice slowly melting, it doesn't really matter if the water is present or drained...as long as there is ice.
When the ice is used up if water is present as heat transfers into the box the water slows the warming. More water slower rise in temperature. The relatively high specific heat and mass of the water require much more heat to change temperature than air or food in the box. That is because the heat flowing into the box and raising the temperature is governed by the mass times the specific heat of the water.
Because the inner wall temperature of the box is very very close to the water and air temperature in the box the difference in the amount of heat transfered by the water compared to air due to the difference in the coefficient of convection is negligable. In other words the temperature gradients are small and the process is not governed by the convective process but rather is governed by conduction through the insulation.
So in conclusion...as long as there is ice in the box it doesn't practically matter if the water is present. In the end game
, when all of the ice is gone then it matters and the water will help to keep the box cold for a longer time.
However, once your ice is consumed you are within a day or so of getting too warm anyway even if the liquid water bought you some hours.
The cooler manufacturers recommend leaving the water in because it is common for coolers to use up all the ice and the cold water will buy some amount of time for picnic'ers and campers on the weekend. In their time frame the water is important.
In a cruisers time scale I think it is less important because a cruiser will need to maintain ice over the long haul. I think the potential heath risks (blood and food soaking in the water, leaking zip locks, etc.) outweight the benefits.