First of all, your repairman is being less than truthful when he says leaving the door open caused the defrost cycle to fail, if those were his exact words. Leaving the door open can cause ice to build up in places it shouldnt, such as the cold air return at the top of the fresh food section. Also, make sure the warm air return at the bottom is not obstructed with anything, inlcuding ice. However, leaving the door open cannot cause a failure in the defrost system. YOur refrigerator needs air circulation from the freezer into the fresh food back into the freezer. If the evaporator coil in the freezer is not encased in frost / ice, check that the evaporator fan is operating, you may need to close the door switch(es), inspect all vents, remove vent covers and thoroughly inspect.

Oh, as a refrigerator repair technician myself, I would not only expect my customer to call me back to correctly fix this, but I would do it at no extra charge since his diagnosis was wrong.