This may be an artifact of where I live, but most rural areas have always had a centralized city (post office, bar, church), and the vast majority of folks who called that village/city their home lived miles away from it on their farms.

Horses and carriages were a practical requirement, since walking all day one way to visit the grocer was unrealistic.

Rural living traditionally meant living in dense villages surrounded by farmland. Lonely farms were more common in the frontier. There was safety in numbers, and living next to other households allowed sharing things that were too expensive for most households.

Not really. At least, never in the US. Plantations, for example, are fairly distant from each other, and from their associated towns.

There are always exceptions and edge cases. Plantations and farms are generally neither known as examples of urban design, nor are they homes for majority of population of any state.

But even using the example of 18th century plantation - the idea of a parking lot for more than a dozen or so horse drawn carriages seems somewhat... straight out of Flinstones.

I agree that You can find some early examples of the modern "carriage-based" urbanism earlier than in the 20th century, but those would be very specific cases. If 90% of the population lives in some denser urban setting and mostly walks, "their" neighborhoods will reflect that in how and where they are built (at least after a generation or two that's necessary for construction). Same reasoning applies if that ratio changes and within a generation or two, everyone suddenly can afford to own a car. The question is now which way do we (and, by proxy, the politicians we elect) want to promote.