This is a great video that touches on your point, the Earth must radiate the heat it receives to stay in equilibrium, even if it uses it to do work in the process.
Everyday objects don't work like that. Saying ice cubes are cold is roughly equivalent to saying that they radiate less heat than the objects around them. If they radiated at "high intensity" then they wouldn't be cold anymore.
"A cold object that radiates heat at high intensity" is a contradiction.