AFAIR,
The "klaxon" in the 350 is for 'HYD' and "RRPM". There is no ENGINE OUT warning based on NG (from memory). You have to determine which problem from the annunciator panel. Should really be treated as a MASTER CAUTION but with limited information.
A "transverse lobotimisation" also helps with maintenance troubleshooting as well.
Completely unrelated to this incident but FWIW - have a look here -
Transport Canada AN
I haven't seen the video and don't know which model the aircraft was, but you can experience some difficulties if a servo accumulator is either "flat" or overpressurised. With a flat accumulator one goes out before the others and they tend to have an argument and the lateral and fore aft are shared across the servo's. Overpressure just gives you less oil to use!
Many have been caught out by the misuse of HYD OFF and HYD TEST. If you have HYD problems the system should be selected OFF without delay as per the RFM.