"Anatole of Paris" was written by Sylvia Fine, Danny Kaye's wife.
She was a lyricist and composer professionally, separately from her husband, but she also worked with him, writing some of his most famous numbers.
I suspect that the "I hate women" conclusion was mostly a joke, based on women's fashion often being ridiculously uncomfortable and expensive. If there is some misogyny in the fashion industry, it is not specific to the film.
One point worth noting is that the 1947 movie takes place in a much broader world.
In the 1939 story, there are brief encounters with a parking lot attendant's horn, for example, but they are as undeveloped as the non-Mitty characters in his fantasies. Even the repeated interruptions by his wife have no real detail, just that it is a nuisance, distracting him from his... well, I would say "rich" fantasy life, but it's not really.
There aren't a lot of details; just brief glimpses of something imagined where he is cool, competent, and admired, but without the details and individualization that would make it "rich."
In the 1947 movie, Mitty has a boss who steals his ideas, a fiancee he doesn't really want, a competitor who does want her, and and overbearing future mother-in-law, all before coincidences set him up for a real adventure.
So the 1939 Walter has five fantasies while doing normal Saturday errands, though as quickly as he falls in and out of them, there will probably be more. The 1947 Walter has some similar fantasies, but also some different ones, and one major real adventure that gets started on his way to work. Then the 2013 Walter has some globe-trotting adventures, but that's not important now.
Watching the 1947 movie it was easy to wonder why Walter was engaged to someone whom he didn't love, but who was also being pursued by someone else whom she seemed to find more exciting. What was in it for either of them?
Yes, you see that under all of the meekness there is courage, but why did he expend the effort to propose to Gertrude? Did she and her mother push him into it before they knew Tubby existed?
That's mainly just set up, and one could assume that 1947 Walter being pressed for help by Rosalind is what saved him from the fate of the 1939 Walter, hating his wife and his humdrum life.
That made me wonder if Thurber hated women.
Based on reading his work, that seemed probable, but reading about his life doesn't indicate it.
Thurber married in his 20s, was encouraged in his career by his wife, cheated on her (but it was mutual), reconciled because she was pregnant, cheated more but apparently had a pretty amicable divorce, then he remarried to a rebound from yet another relationship (though that one was still in contact with him years later). You could argue that behavior is not characteristic of loving and respecting women, but it's certainly not avoiding them.
To be fair, some of his work is not merely misogynistic, but also fairly misanthropic, and yet he had lots of friends. Later health issues led to emotional instability and depression and he tended to drink too much. His work did get darker then, but it seems like it was dark all along.
I have noted that the women he portrays -- in cartoons and in prose -- tend to be large and domineering. There could be an aspect of resenting the influence that women had, even though it seemed to involve things that were helpful, like pursuing writing as a profession and caring for him through his health problems, as well as picking up after him emotionally.
However, there is also a part of me that wonders if it was just going along with the ease of complaining about wives. Apparently, Henny Youngman adored Sadie, even if you would never know it from his act:
https://www.cracked.com/article_41738_take-my-wife-jokes-began-with-a-genuine-request.html
I also wonder if part of it is when the woman commits the crime of becoming a wife, with all of the interest focused on obtaining her, followed by disappointment in her being a real person instead of a fantasy.
The fantasy seems to be based on a rather simplistic definition of masculinity and a lot of other nonsense, but then at least it becomes convenient to blame your perceived problems on your wife as opposed to other social forces that would require some level of cooperation motivated by more than self-interest.
Better still than blaming yourself.
Of course, Thurber says that Mitty was based on Robert Benchley, but I'm not sure the attitude fits. Maybe Benchley actually had a rich fantasy life.
The IMDb trivia page for the film also says that Thurber offered Samuel Goldwyn $10,000 not to make the film. He did not like the film version, as it wasn't his vision, but I don't think anyone would have enjoyed a film of his vision.
That's show biz.