Switch off with a bag

Just experimented using Grommits transmitter. Bottom line, it's my transmitter alone that has the fault.

Does the X9E auto switch-off when the receiver is still switched on?

Yes

Looking at this picture I would guess that the power for this unit is controlled from the momentary action microswitch in the middle of the board. I would further hazard a guess that the line from this momentary switch to the real (silicon) power switch is floating (i.e. it should be tied to to a power rail by a resister to avoid triggering by electrical noise, but isn't). Normally there would also be de-bouncing circuitry to ensure accidental knocks and vibrations and static electricity inputs are ignored, if this is not working either I would suspect that there is either a missing or loose resistor or a dry joint or broken track in this area.