There really is no iceman conspiracy, and a lot of us grew up boating with block ice coolers for good reason. The ability of ice to keep something cold is hugely dependent on the mass of the ice. Relatively speaking it takes very little heat to warm a mass of ice 1 degree, but it takes A LOT of heat to warm the same mass of ice from frozen to water. The huge heat absorption during this change of phase (solld - liquid) is what makes ice work so well. There is very little value in supercooling ice, much more value in stripping the heat from your food and drink - pre-cool them so they have less heat to contribute.
Putting some numbers to this.. (which I think is all good advice)
The numbers for the energy to change the temperature of water per weight for for the different states are approximately
Ice is 0.5 cal/gram C
Ice to water 80 cal/gram ("heat of fusion")
Water 1 cal/gram C
So lets say you had 1 gram of ice that starts at a typical freezer temp of 5F and look at what energy is involed in this one gram going up to about the warmest you want a cooler at 45F.
First, the temp needs to be converted to C so we start at -15C to 0C as ice then 0C to 7.22C as water
For this 1 gram,
Ice state calories = 0.5 * 15C = 7.5 calories
Ice to water = 80 calories
Water calories = 1 * 7.22 = 7.22 calories
As Gunni said, the majority of the energy absorption happens in the ice to water transition.
I made an ice chest for my trailer sailer that has an easy to use water drain at the bottom plus there is an about 1 inch high platform that the food sits on. Sometimes I put frozen jugs in the cooler but sometimes also bags of ice. If I put in bags of ice, I will drain the water out usually in the morning of each day so I never have "food floating around in a water ice mixture" that we all like so well.
According to the numbers, going from out of the freezer to water, I get the benifit of 87.5 cal/gram (ice temp change plus the transition from ice to water). But I throw out the ice melt so Im throwing away 7.22 cal/gram.
With what Im doing (empty the ice melt each day), Im loosing about 8.25% of the cooling benefit. But I never have food floating around in some funky colored water.