AFAIK leaving the tank open to drain is a bad idea, at least with black tanks, so I suppose it is the same with the galley grey tank.
IIRC you are supposed to let it build up some, so when you do open the valve there is enough there to do a real dump and clear it all out.
Our kitchen grey readings are usually higher than it really is with gunk on the sensors, but with yours being actually plugged up, you need either a physical poke at it, or some kind of Draino.
The ice cube trick only works if you can get ice cubes down the drain (by-passing the sink connection underneath?) and you can drive around so the ice cubes can do their work.