So they had been running fine for 200+ years by us; what happened?
The Brits have had those contracts for years. Most ports are owned by local Port Authorities. The Port Authority has a lot of varied work (some of them are also responsible for airports). The largest ports have a lot of the routine work contracted out, and container transfer is just one of those things.
I don't know why no American co. has ever gone into that business, but the truth is that none did. Internationally, there seems to be only 3 cos. left who do it - this one, based in Dubai, and one each in Singapore and Hong Kong.