There seems to be a little delay when switch off until current drop to 0,so some part of soft on-off seems to work,at least for off part.
Most probably is something on power switch taht provide not enough power to mcu or even mcu hang on boot sequence
Maybe power switched by rotary isn't all that is required to power on.
I don't own this model so I can't test.
With no batteries switch rotary to some position and insert battery,is it powering on?
If in sleep mode switching rotary to another position wake it from sleep?
There may be some tracks on rotary switch that send some pulse during movement to mcu that may be required to reset sleep timer counter but also maybe required at power up.
Can't remember from older video but I think programmer cable powered multimeter even it was in off position,so simple check just by connecting cable to PC.