The decline of formal wear, notably the suit, and most significamtly of hats (everyone wore hats during the first half of the XXth century, which is something that movies strangely seldom portray), is something that happened during the sixties.

In fashion history, the 60s is often portrayed as the triumph of "youth culture". It is a period marked by the growth of consumer culture and by a decline of traditional and moralistic notions of what dress is appropriate in public. (Twigg, 2013)

There was also a rise of counter-cultural movements that adopted very definite styles of dress that were in rupture with traditional dress. Dick Hebdige wrote a lot on british youth subcultures, such as Mods, Rockers and Punks, and the way that they respectively influenced mainstream fashion. In the US, one simply has to think of Hippies and their rejection of everything associated with conformity, consumerism and traditional values. You may think of these groups as marginal, but they durably influenced the way that everyday people dressed by breaking the mold of a singular acceptable fashion style.

Before the 60s, fashion was mainly "trickle down": most people at various stages of wealth tried to imitate the fashions that the wealthy would wear (in most cases, this meant things coming from Paris, although the City of Light temporarily lost its place because of the german occupation in WW2). Youth subcultures changed that because trends now emerged from lower socioeconomic groups as well. If you look at some designers from this period (I'm thinking Giorgio di Sant'Angelo), there was a clear inspiration from hippie styles in high fashion. In the same vein, Vivienne Westwood, who was a prominent figure in the emergence of punk fashion, later on became a world-renowned high fashion designer. Fashion was now something that could emerge from anywhere, which gave way for the development of personal style that we know today.

This is my answer, but I know that this sub has very talented fashion historians that may provide additional insights to your question. Perhaps u/mimicofmodes has something to add?

References

Breward, Christopher. Fashion. London : OUP Oxford, 2003.

Hebdige, Dick. Subculture: The Meaning of Style. London: Methuen, 1979. -> This is the book I recommend you consult if you want to learn more about this topic, but take note that it focuses on the UK rather than the US.

Twigg, Julia. « Fashion, the Body and Age » , in Black, Sandy et al. The Handbook of Fashion Studies, London : Bloomsbury Publishing, 2013.

Welters, Linda & Littlethun, Abby. The Fashion Reader. 2nd edition. London : Bloomsbury, 2011.

You've received an answer, but as I've been Summoned I will also leave you with a c/p'd older one of mine:

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.

My go-to source on the overall casualization of dress in the twentieth century is Dress Casual: How College Students Redefined American Style, by Dierdre Clemente; if you're interested in the subject, you may want to find a copy of the book.