Logically when you then are creating media where cars don't exist, you're going to need to create an equivalent and it's "common knowledge" that horse carriages were the big thing cars replaced. So you instead get a horse, something that whilst obviously much slower than a car, still gives the impression of "this lets you move faster" than how a human would walk on foot.
There's also the image of the most famous start/stop horse network of all: the Pony Express, which probably played a part in setting up this imagery - for an onlooker, postal workers would travel from one side to the other on a single horse, permitting (for the time) very fast deliveries. In reality it was a series of well maintained stables and lots of start-stop segments to keep a horse in good condition, but that's not something you're going to notice if you're not a mailman.
(The Pony Express also wasn't sustainable and went bankrupt rather quickly, but it's legacy is pretty lasting.)