I have never seen AP staff pack goods in boxes/bags for customers. The most the PO's I go to do is put extra tape on boxes if they don't look secure enough.
The staff would never get through the lines of customers they have throughout the day if they were responsible for packing customers parcels.
Some staff would have no idea how to pack a parcel.. They are employed as customer service officers to charge people the correct postage costs, check passport applications.. They don't have parcel packing training.

How would the sorting and delivery people along your parcels route even know who packed the parcel?