Not quite. Mail handlers are involved in every stage of transport EXCEPT at the sorting centers themselves. How do the packages and letters get TO the machine? How do they get from the machine to the truck? And what happens when the package is at the post office itself?
My post office has no DPS machine (many individual offices don't). We rely on the main Atlanta sorting facility to not only sort it by post office, but by routes within the individual office too. And the main Atlanta sorting facility is about 40 miles away. That's a long way for sorted mail to travel and somehow stay in order.
Packages don't get sorted by route for us. At the main sorting facility, they only get divided up by zip code (post office). They arrive shrink-wrapped on wooden pallets which are brought inside one by one. One or two people stand in the middle of the back room surrounded by about 15 rolling bins. They take the package off of the pallet, determine which route it goes to, and toss it in the bin for that route.
The actual sorting machine is typically only used twice. First is to determine which flight it goes on (if any) when being sent out of the area. The second is when it arrives in the area where it will be delivered, and gets sorted to office/route. For example, I ordered chicken diapers recently, sent from Minnesota. They went from the lady's house, to her post office on a carrier truck. From there someone put it on an outgoing pallet and shrink-wrapped it and loaded it on to a truck. It went to the sorting facility in her area on this large truck, went through the machine, and was assigned a flight. Another truck took it to the airport where it was loaded on to that assigned flight. It flew to Atlanta and got picked up by a truck and brought to the sorting facility. There it went through the machine again and was sorted to my post office. That determined which pallet and truck it went on to be delivered to us. But again, people stand in the back room and sort the packages by hand once it gets to our office. Meanwhile, it's getting bounced around on hand trucks, fork lifts, pallet jacks, in people's hands, and don't forget the person actually tossing it into the bins at the actual post office.
Indeed machines replace a lot of the work that used to be involved in sorting packages. But they are still VERY far from doing all of it. And if one person can see the "FRAGILE: DO NOT SHAKE" notice on the side of the box and HAND-CARRY it to the machines or the route bins, the pallet for that Post Office or where ever, then it can go a VERY long way in stopping a lot of the sharp jolts these packages take.