Nowadays we have container ships. But once upon a time cargo ships would swing out their davits and haul cargo aboard in nets – resulting in much comedy from the likes of Abbott & Costello and Hope & Crosby.
Is the old cargo net system still used anywhere?
Not in the way you think.
Nets these days are used to keep cargo in place, not sling it around. Loadmasters in the military use them extensively on all vehicles, and in civilian use they are used to keep loose items from flying out of trucks.
Ammunition transfer operations between ships. Also,still used in break-bulk situations. Used to transfer blocking and bracing materials (timber, strapping, tools) into and out of the holds.
While not so much break-bulk in modern ports, you will see more in less-developed nations. Also damaged containers/pallets will often be removed with a cargo net.
Many ports have a substantial vehicle fleet of mobile cranes to carry out smaller operations on non-containerized cargo.
Expanding on this, I saw a video on youtube awhile back (can’t find it now) showing the cargo and materiel transfer between a supply ship and an aircraft carrier. Everything was transferred in nets IIRC. Not exactly what the OP was looking for, I suppose.
I’ve seen it in action when I had a job doing audits on board cruise ships - granted, it was a pallet stacked with cardboard cartons, but they dropped it onto a big cargo net with a fork lift truck, hooked the corners and edges of the net to a great big hook and hauled the whole lot up with a crane, onto the deck, then manhandled the cartons into the storeroom (I helped).
I can’t remember where it was though. Might have been Barbados.