Would huge ice cubes work at cooling better than water? Just wondering....
Very good. You are correct. Ice, as it melts, absorbs more heat because you are taking advantage of the phase change. This is called latent heat of fusion, and involves the concepts of enthalpy of fusion and latent heat. For ice, the heat of fusion is about 334 kJ/kg. Whatever the heat source is that melts the ice "gives up" 334 kJ of thermal energy to melt 1 kg of ice. That makes whatever is supplying the energy cooler.
In this case, they wanted to transport coolant through a structure with complex geometry, so liquid was the best choice. Large chunks of ice would have a hard time being transported through small channels and openings.
You can get added heat removal through the phase change from liquid to vapor. This is called the latent heat of evaporation. It is the principle upon which cooling towers work. The steam you see coming out of a cooling tower is removing heat from the power plant through this process. BTW, in a nuclear plant, the steam coming from a cooling tower isn't from the reactor. The coolant running through the cooling tower is from the condenser recirculating loop. What is being evaporated is from the heat sink, typically a lake (natural or man-made).