It's a complex issue, but I see American car culture as much more recent and derived from suburbia, rather than from rural farm life. Very few people in the era when the majority of Americans were farmers had cars, because they were very expensive, and patterns of daily life didn't involve driving the kinds of distances to big freeway strip malls that you see today (neither freeways nor strip malls existed). The family may have had a tractor which was sometimes used to drive into town or a bigger city (or a horse was used instead), but driving daily to the Super Wal-Mart is a post-urbanization trend.

It's anecdotal, but certainly none of the farming background in my family involved car culture. My great-grandfather lived in a small farming town in Kansas, and they didn't own a car. The town, despite being small, was very centralized around a compact "main street" where all the shops were in walking distance of each other, pretty typical of any Midwestern town prior to WW2. People who lived outside of town either rode into town on a horse, took a tractor in, hitched a ride in, or walked. If they needed to go longer distances, they used this method to get to the nearest town with a train station, and took the train. They bought a car when? When they moved to Los Angeles in the 1930s. The rise of car culture and suburbia also led to most of these traditional compact farm towns turning into what you see today, with old downtowns are that are abandoned or semi-abandoned, replaced by a string of strip malls on the bypass freeway—a bypass freeway that in most cases was built in the 1950s or '60s, well after most Americans lived in cities.