If Airbus bloke is referring the Airbus 320 family it is indeed due to the crossbreed valve staying closed. As to why, looking at the procedure logic indicates the prevention of cross-contamination.
If the engine fire push button was not pushed, which means there was no damage or fire, then there is no problem running wing AND engine anti-ice from the live engine. Only limitation is that one pack needs to be switched off when using wing anti-ice!