I absolutely agree about cars, but in what way is it "difficult to live without" a big home for the average American? How does zoning make that the case?
If zoning rules allowed developers to build larger, taller, and more numerous buildings, we would all have better, cheaper, and more transportationally efficient housing.
Zoning as well.
2. They're not being built where the jobs are. Take the SF Bay Area and the SV jobs/housing crisis. Much the problem is that many of the jobs (still) are on the Penninsula and South Bay, while high-density housing (and only somewhat so at that) is in San Francisco. Santa Clara and San Mateo counties are largely suburban sprawl, with very few exceptions. Transit stinks (virtually no usable busses, little rail other than CalTrain and VTA), and even riding a bicycle is a dicey proposition with traffic densities and poor bike lanes.
3. Schools. In the US, housing choices, especially for families, are driven by school access. Good schools are typically suburban. E.g., sprawl.
4. Urban blight. Less an issue now than previously as there's a flight back to cities, but the dense early housing belts around older US cities build in the period 1900 - 1945, that is, before the automobile was widely used, tended to be less attractive due to crime, poor schools, and, well, the attraction of a cheaper, brighter, cleaner outer-suburban belt. The point remains that denser housing stock is also quite often older housing stock.
5. Building codes. Various factors, including safety, appliances, access (doors, steps, stairs, etc.), have tended to make homes larger. One of the reasons the "tiny homes" movement has a preference for building on trailer platforms is that trailer and mobile home requirements are far less stringent.
6. Lending availability. Again, for various reasons, financing is more available for newer, larger, and suburban homes (additional reasons including redlining and government policies from local to national levels have also existed historically). Older and smaller homes are harder to buy, and to sell.
7. Builder and architect familiarity. There's an existing workforce and skillset around the "conventional" product. Building experimental, high-density, high-efficiency, or exotic (often, ironically, highly local or available) material homes pushes you to odd skillsets. It's a bit like basing a tech startup on a little-used programming language or operating system.
8. Major builders and planning departments. The "easy" direction for major builders, and for city and county planning departments, is what they know and what they know will sell. Given that a major builder might easily account for a substantial amount of all home construction nationwide, that's not an inconsiderable concern.
The upshot is a lot of structural obstacles to constructing a widely different housing stock, particularly doing so throughout an entire town (let alone a city), which is what would be required to bring up overall densities. One household building one small home won't make the difference. You'd have to change an entire community's mindset. That's ... challenging.