After reading the wording carefully, I believe the question and the OP are asking, during melting, how does the temperature of system change (if any)? The question is not asking what happened to melting point of ice.
The temperature decreases is simply because melting of ice is endothermic (and assuming the system is adiabatic so the energy of system but not of surrounding is used for melting). It melts because the system is moving towards most thermodynamically favourable composition.
Considering pressure at 1 atm,
if you consider melting of pure ice at constant temperature (isothermal), the melting is endothermic too but the net effect is that energy used for melting is taken from surrounding not from system, so the temperature of system does not change, all ices melt into liquid water if that constant temperature > 0 °C;
else if you consider melting of pure ice adiabatically (no heat enter nor leaving the system), the temperature of system will drop as melting goes because the system uses its own energy for melting, until equilibrium temperature (0 °C) reached, and then the melting stop. Solid and liquid present at equilibrium, not all ices melt;
else if you consider the melting of ice in the presence of salt isothermally, melting happens if that constant temperature > -21.1 °C.1 Why -21.1 °C but not 0 °C ? That is a thermodynamic effect due to mixing, and apparently this effect is more significant than colligative freezing point depression. Of course the system temperature will not change, same reasoning as (1);
else if you consider the melting of ice in the presence of salt adiabatically, the temperature will drop to -21.1 °C as melting goes, same reasoning as (2).
I think the question is asking for (4), and you are confused because treating (4) as (3).
1 Peter Atkins, Julio de Paula. Physical Chemistry (8th Edition). OUP. 2006. Page 189, (a) Eutectics