Bullshit. If I have a glass half full of boiling water, and a glass half full of ice water, the two glasses have an average temperature of around 50 degrees C. If I pour one into the other, the hot water will cool, and the ice water will warm; but the average temperature is still 50 degrees.
The heat was redistributed, but the average temperature hasn't changed.
No, in your example the average temperature will drop but the total heat of the system will remain the same.
In the commonly understood meaning of ice water, you will have a mixture of ice and water. Such a mixture is understood to have a temperature of 0 degrees centigrade but additional heat needs to be lost to make the transition from liquid water at 0 degrees C to solid water at 0 degrees C (heat of fusion). The mixture of your water at the boiling point and your ice water will equilibrate at a temperature below 50 degrees C. The actual temperature will depend the percentage of water that is in the form of ice in the ice water.
So if the heat can go somewhere other than to change the temperature of water you can have changes of the mean temperature of the water.
Likewise in the rest of the comment, a global energy balance surplus need not mean a short term global surface temperature increase and an energy balance deficit need not mean a short term surface temperature decrease because the energy balance affects more than air temperatures. The oceans, in fact, act as a massive heat sink (and the data is there showing that surplus heat is going there). That is, additional heat affects more than surface air temperatures, it affects ocean temperatures. As a result, anything which affects the heat balance into this heat sink will affect air temperatures. This means that a heat surplus can be masked if additional heat gets temporarily dumped into the ocean but it also means that if the process that is dumping surplus heat into the ocean decreases you will see an atmospheric temperature rise.