Classic Man vs Technology stories like that do have a bit of a deeper theme. Afterall, John Henry is wearing clothes likely spun from a machine and wielding a tool, if a primitive one. He's not punching the stakes into the ground with his fists. The important bit is that machine he faces is automated. Something built and constructed using math and knowledge far removed from the day to day experiences of the working man.
It's just as much about the man on the ground versus the pencil pusher out of town. The experience of a craftsman, as lowly as that craft may be, versus the generalized, interchangeable automation. A similar story could be spun out of a man who built him self custom robo parts and who spent his whole life tinkering with his own machines for specific tasks and a general automaton made by a committee across the country.
It probably wouldn't be as drammatic of a story. The story does rely on a more contrasting image of a man and a hammer and a complicated machine of industry.