Why do we cool water with ice? I've came up with the following question: Why do we cool water with ice rather than with steel given that the last one has a bigger thermal conductivity?
 A: Because ice is (usually) at a lower temperature than steel. It also absorbs energy when it melts.
You can calculate the amount of energy absorbed by either steel/ice with $mc \Delta T$. With ice you add an extra amount due to it melting. This yields the maximum amount of energy that can be absorbed. If you have enough steel at sufficiently low temperatures, you can use it instead of ice.
A: There are several factors (temperature, specific heat, volumetric heat capacity, thermal conductivity, and latent heat of fusion for ice) to consider. See the data below for steel and ice. Their relative importance depends on exactly what your goal is (lowest final water temperature and/or time to reach that temperature), how much water, ice, and steel you have, and what the initial temperature of each is.
For example, if you are starting with the same volume of ice and steel at the same initial temperature, then before the ice begins to melt you can cool the water, and more of it, faster with the steel than ice. That's because both the volumetric heat capacity ($C=\rho c$) and thermal conductivity ($k$) of steel is greater than ice. That means the rate of heat transfer and the amount of heat that can be absorbed by the steel is greater than can be absorbed by an equal volume of ice.
On the other hand, if instead it is the initial mass of steel and ice that is the same, or if the ice melts, then initial amount of water may determine the answer because the specific heat of ice is almost 5 times that of steel and the heat that can be absorbed by the ice is the large latent heat of fusion.
Hope this helps.
For steel
c=0.461 KJ/kg$^o$C
$\rho$ = 7500 kg/m$^3$
$C$=3458 kJ/m$^3$
$k$= 45.3 W/m K
For ice
c=2.03 KJ/kg$^o$C
$\rho$ = 917 kg/m$^3$
$C$=1862 kJ/m$^3$
$k$ = 2.22 W/m K
$h_F$ = 334 kJ/kg = 306,278 kJ/m$^3$
A: Because the ice melts and the phase change from solid to liquid enhances the cooling of the liquid due to the energy of phase change (energy removed from the liquid to melt the ice).
