I think the most important thing is to point out that the car concept is much older than the first car in the modern sense. The first rail-less vehicle for human transportation was a rudimentary steam car, in 18th century France. So someone had the idea, before trains were widespread. So, why did it not catch on like trains did?

The important thing here is that the propulsion technology in early trains (and that prototype car) is steam. If you don't know, steam engines work by boiling water, and expanding the resulting steam in cylinders to produce useful work. This is how steam locomotives work (until they started getting phased out in the 20th century), unlike cars that use the expansion of burning gasoline/air. (There have been steam cars, most famously the Stanley Steamer, but widespread use of cars was mostly using internal combustion engines).

The steam locomotive model isn't readily scaleable down to a car. Keeping the locomotive steaming properly is not a trivia task. 2 people are required for this in most locomotives. They're also fairly maintenance intensive. Early cars weren't either, but there's extra complexity there that makes it difficult to use. This was an issue with steam cars, too, and that was when steam locomotives were a decades-old technology (though they did get ok at the control complexity thing). This made the propulsion technology of the day not practical for cars.