I've started seeing things about the current situation changing car culture with remote work becoming more normalized. Wondering if that last big pandemic had effects at the dawn of the automobile. (The first Model T was produced in 1908.)
Short answer:
Roughly the same time, but largely unrelated.
Discussion:
This is not a bad time to discuss the commonplace that "correlation is not causation"; that two things happen roughly around the same time is not evidence that they're linked.
If you want to stretch a bit, you can identify the epidemic with the idea that cities were unhealthy and the growth of the suburb-- but suburbia rose long before automobiles, so you can't make "car culture" the gating function. The notion of the unhealthfulness of cities is a commonplace of the 19th century -- New York has a cholera outbreak in 1892, and tuberculosis was common. These were far more persistent threats than influenza, and do contribute to the growth of suburbs . . . but these are suburbs serviced by train or steetcar, long before automobiles are widespread-- though automobiles do _fit_ this new suburban lifestyle.
The 1920s do see explosive growth in automobiles -- 8 million registrations in 1920, with 24 million by 1930-- but this is essentially all after the pandemic had passed. It's hard to argue that someone buying a car in 1926 was doing so because of something that had happened in 1919. Far more influential would be legislation like the Federal Highway Act of 1921, which provided Federal matching funds to help the states build modern roads.
US Government data shows growth in automobiles per capita over the years-- the upward trend of the 1910s and 20s is essentially constant, showing no evidence of any change in the years that were affected by the influenza.
Sources:
Ueda, Reed. “The High School and Social Mobility in a Streetcar Suburb: Somerville, Massachusetts, 1870-1910.” The Journal of Interdisciplinary History, vol. 14, no. 4, 1984, pp. 751–771.
Selwood, H. John. “URBAN DEVELOPMENT AND THE STREETCAR: THE CASE OF WINNIPEG, 1881-1913.” Urban History Review / Revue D'histoire Urbaine, vol. 6, no. 3, 1978, pp. 34–41.
Oak Ridge National Laboratory, Transportation Energy Data Book: Edition 35, ORNL-6990, Oak Ridge, TN, September 2016, Table 3.7.
MARKEL, HOWARD. “‘Knocking out the Cholera’: Cholera, Class, and Quarantines in New York City, 1892.” Bulletin of the History of Medicine, vol. 69, no. 3, 1995, pp. 420–457.