Don't exactly understand what you were trying to say there at the end. But, indeed:
The enthalpy of fusion is the amount of energy required to convert one mole of solid into liquid. For example, when melting 1 kg of ice (at 0 °C [...]), 333.55 kJ [about 80 (kilo)calories] of energy is absorbed with no temperature change.-Wikipedia
Thus, 1 kg of ice at 32 °F has a far greater cooling power than that same 1 kg of H2O at 32 °F in liquid form. Indeed, merely melting 1 kg of ice - without any temperature increase - would absorb as much heat from the surrounding environment (e.g., the burnt flesh) as would be required to then proceed to elevate the temperature of the resultant meltwater another 80 degrees Celsius (approaching boiling)!
If you want to cool a burn wound, ice would thus be more effective than ice-water - though solid ice would display little to no convection (one of the three ways heat can be transferred). In other words: Liquid water can transfer the heat away from the wound faster.
Light, loose powdered snow would probably be a better choice, but nevertheless, I think that I'd prefer ice-water - since the objective isn't to freeze the damaged tissue, but rather just to remove the excess heat left over from the actual burn event.
Once the burned tissue has been brought back to about 98.6 °F, there's no curative benefit to further cooling. The only benefit further cooling would provide would be to numb the pain (not to be underestimated!).
Regards,
The only reason ice water is the better option is that it’s less likely to cause frostbite.