What about the conductive path of the H2O inside the hairdryer motor and switch? Surely the motor and/or switch is connected tothe neutral and ground, right?. Well if there is water inside the motor, the GFCI *should* trip - with or without a grounded sink or water basin.
The GFCI operates by sensing the difference between the currents in the Hot and Neutral conductors. Under normal conditions, these should be equal. However, if someone touches the Hot and a Ground such as a plumbing fixture or they are standing in water, these currents will not be equal as the path is to Ground - a ground fault - and not to the Neutral. This might also occur if a short circuit developed inside an ungrounded appliance (i.e. water bridging hot and neutral surfaces) or if someone was working on a live circuit and accidentally touched a live wire.
A GFCI tests for a Grounded Neutral Fault and a Hot to Ground Fault. Both of these should occur when an appliance is dropped into water - thus tripping the GFCI circuit. The video appears to show a defective GFCI plug unit. As to why the GFCI receptacle did not trip, I think it has something to do with the defective GFCI plug. I wonder if the GFCI receptacle would properly trip if you cut that GFCI plug off and installed a standard NEMA plug?
One other check would be to see if the GFCI receptacle was correctly wired. I have seen retrofit installations where the installer 'tricked' the GFCI by jumping the neutral and ground on the GFCI.
Just my $.02...