Nope ... don't buy it.
Asbestos is not a conductor and the insulating jacket that surrounds the cable is not conductive either.
The only way I can see him doing this is if the wiring is BX. Then he would be relying on the metal jacket of the BX being the ground conductor back to the panel. That would be assuming the BX is secured to each box properly with metal strains.
That's a pretty big assumption the jacket is in good shape ... maybe he has X-Ray vision....he mentioned something about the old (in good shape) casings/insulation/protective fabric which he says is made from asbestos?? That he can get a grounding off of that????