Start with a maximum of 1150 W/m^2 input power from the sun. Whatever is absorbed will raise the temperature of the material doing the absorbing. If there were no way to remove the heat, the material would continue to increase in temperature until it was able to emit 1150 W/m^2, by whatever means possible.
Now, it were the same temperature as the environment, there would be no emission of heat, clearly undesirable.
Therefore, in order to remove the heat, the material MUST be hotter than its environment.
Likewise, conduction into the house requires that the wall be hotter than the interior of the house.
Stucco, being not very smooth may have a poor convection coefficient. Possibly, it may have a lower thermal conductivity and thus keep more heat near the surface.
TTFN