I think what this ultimately comes down to is that in western history up to about 1350, women were significantly more concealed than men by their clothing; from about 1350 to 1500ish we see a transition where both men and women show much more of the body than before; and then from about 1500 on, women are consistently more "revealed" than men.

The reason for this is that concealing/revealing don't just relate to the skin, but the body itself. From the time of the "tailoring revolution" on, women's fashion has focused on the figure, despite the frequent use of long sleeves and long skirts: the sixteenth century is when boned or tightly corded underbodices seem to have become standard wear, at least in elite/genteel contexts, and their use was highlighted by an outer garment that clung closely enough to show that one was wearing a foundation underneath. We see this continuously in the following centuries; even in the Regency, when the focus was no longer on the waist, the bodice was still tight in the bust to show the use of the fashionably supportive foundation. (At the same time, showing skin was still associated with women's fashion to some degree: the invention of evening dress in the nineteenth century led to a standard where women were sleeveless and wore lower necklines in formal eveningwear.) And in addition to the desire to be fashionable, women faced an expectation to be attractive - to show off their narrow waists and other body parts.

I have a few past answers relating to this:

What led to the changes in tailoring and fashion in 13th Century Western Europe?

How often did women replace their corsets or stays?

How did the corset come back to style?

during the victoria era, why was off the shoulder fashion invented?

But why the change? Well, I have written extensively on the casualization of fashion in the twentieth century, so I will copy and paste:

Over the course of the twentieth century, the increasing availability of college to the middle class led to "college culture" becoming more defined and shared by more people. While hats continued to be seen as normal aspects of campus dress for female and male students through the 1930s, during and after World War II, youth culture began to regard them as unnecessary and conservative. (Other key features of casual college dress were jeans, especially on women; shorts for men and women; the oversized women's cardigan, called the "sloppy joe"; flannel shirts and lumberjack coats; sweatshirts; and tennis shoes.) When hats, suits, and more formal dress were worn on college campuses, it was because they were mandated by dress codes in the stricter and more formal schools, like Radcliffe.

As the students who were used to dressing this way grew up, their values eventually became the new ordinary middle class values - of course many conformed to the standards of "adult dress" when they graduated, but these changes never happen overnight. Eventually, there were enough adults hanging onto the college style that it became acceptable to bring many of these items into the casual and, in some cases (such as hats), even professional wardrobes outside of college campuses. In addition, going forward into the 1960s, you have the hippie movement and its co-option into everyday fashion (which again related to youth culture on college campuses): rejecting the professional, suited norms of the 1950s even if you were not actually going to wear a poncho and live on a commune.

As you've noticed, there is a Great Man argument regarding Kennedy just deciding not to wear hats and the nation - heck, the globe - following suit, but it's clearly more a symptom rather than a cause. Some also make practical, common-sense arguments regarding the height of a car ceiling or less need for protection from the elements, but I find them unlikely. Good-quality felt homburgs and fedoras had not been worn by businessmen to protect their heads, but because it was accepted that You Wore A Hat - it was fashion, not function, and although everyone likes function-related arguments for fashion changes, there is generally little evidence for them. If men had continued to wear fashionable hats, car manufacturers would likely have continued to leave enough headroom for them.

Really, you also have to take into account the fact that the dominant ethos over the twentieth century, except in certain subcultures, not only prioritized convenience but treated it as the most "natural" and rational reason behind just about everything. Wearing fewer clothes with simpler closures and a looser fit and cutting down on accessories was part of that, as was the increase in canned or packaged foods, greater amounts of disposable goods and tools, etc. I don't want to get too far outside my field here, but just as the phasing out of the hat is part of a wider change in fashion, the change in fashion is itself part of a wider cultural shift.

Part of this shift to being more casual involved the restrictions on women showing their bare arms, legs, or decolletages being loosened. As noted above, women have faced (and do face) an expectation to be attractive in everyday life, which has resulted in cultural leniency in showing more skin.