Nice analysis. I love this kind of analysis of social phenomenon. But I will add that not only do cars make hats inconvenient they also make them less necessary. You spend less time standing around or walking around outside when you drive, especially if you have a garage. So you spend less time exposed to the elements, like sun, wind and cold. I will suggest this might help explain why hats were less common for women: They were more likely to be homemakers or do "pink collar" jobs. So they probably generally spent less time outdoors than men.

We still have men who wear hats at work but we typically don't mistakenly think of it as a fashion choice because they tend to be hard hats or otherwise clearly utilitarian. Much of what we view as mere fashion has either a utilitarian purpose or a utilitarian origin that has since been forgotten.