They could've worked in cities. Lead-acid batteries, though, have never had the energy/weight ratio necessary for the type of long-range driving necessary outside of highly urbanized areas.
Two of my great grandfathers bought cars around 1915. One owned a plantation outside of Memphis. And the other a ranch in California. They didn't have electricity.
Not developed by my comment is in 1920 you didn't 'need' a car in cities. Since you had street cars. In rural/small town America cars were really useful but electric cars were a non starter.
I would say that fossil fuels are very important for a lot of our battery technology. I believe that a large number of important chemicals and materials like plastics are derived from oil.
They are important, but not as important as they are to transportation. The quantities of oil required for fuel production dwarf the quantities required for plastics and other important chemicals. It's still a sizeable amount but not even close to what gets burned every day.
It's just that fossil oil is - relatively speaking - very cheap compared to other sources.
