Just a rough guess on this might be the rear belt alignment.
If the belt alignment is out, it would always run off to the same side of the pulleys. When backing up the bike by foot, the belt will be running to the opposite side.
Sometimes at very slow speeds, while pushing by foot, the resistance is such that it will resist the foot power. Under power of the motor, it will probably just start wearing the side of the belt against the rear pulley. I've only seen one drive pulley and it was way wider than the belt so I'm guessing any guiding the belt might need is done by the driven rear pulley.
I know that on timing belts at work, sometimes if out of alignment, they will rotate by hand one direction but lock up in the other. When powered up they just run and wear the belt.
I'd look at the belt for excessive wear or black powder on the rear pulley or the area exiting the front trans drive pulley area. You may also see a VERY shinny side of the rear pulley compared to the opposite side of the pulley guides.
Good luck, but get this looked at quickly.