Ice works two ways to keep your stuff cold. Both processes absorb heat.
First, let's stay you start with very cold solid ice from a deep freeze -- between zero and -10 degrees F. Let's also say you were smart and stuck a few blocks in your cooler a few days before as sacrificial ice to pre-chill the interior of the cooler. That ice melts faster.... Once your cooler is packed, you head out into the wide world that's above freezing. Heat makes its way into the cooler. The ice absorbs that heat and the ice warms up, but doesn't melt. This makes sense, and yes, frozen food would do the same thing.
The next part is the magic. It involves phase change and latent heat. Once the ice warms to the melting point and is isothermal (all the same temperature), it continues to absorb heat, but it does not change temperature. The heat energy is needed for the phase change. As the ice was warming from -10 to 32 degrees, that was "sensible heat," so named because you can sense it. Latent heat doesn't change the temperature. Water is such an awesome chemical, I wish there were some way we could use it for recreation....
Once the ice is melted, it's still 32 degrees. It still can absorb more heat, and this is again sensible heat. You still have safe food until the temperature gets to 41 degrees, and that's when you enter the danger zone.
Here's a quiz: Can you use something else frozen that will work like water ice that isn't water? I use frozen BEER as part of my ice. It still goes through a phase change, but does so at a slightly lower temperature because it's a solution of sugar and alcohol in water. But it's still absorbing heat as it melts, so the water ice won't need to absorb as much.
I use frozen jugs of water to keep my food dry, so I don't drain. The real question is sodium hypochlorite, chlorine dioxide, or quaternary ammonium.