It was a recurring theme throughout most of Golden Age fiction.
E.g. Clifford D. Simak mentions them as a mode of transportation in The Goblin Reservation, Asimov has them in Robots of Dawn, and I'm sure I'm forgetting plenty more.
It could be that it was Heinlein who kicked of the trend.
I believe it was H.G. Wells, in his A Story of the Days to Come (1897) and When the Sleeper Wakes (1899).
Back in the heyday there was this idea of the arcology where a group of people had it with their government and made their own city-utopia which would rule itself.
Very often in those they featured technology like the staggered automated walkways for transporting people around, etc.
People have tried faster moving walkways many times. The problem is getting humans on and off such a system safely in a way that is easy to maintain.
In practice, everyday transportation systems need to accommodate a wide variety of users safely, like a toddler, or a commuter holding a cup of coffee, or a grandmother with a walker.