Am getting around to chipping in my two cents after lurking in this thread for a while.
For me as well, “Father Knows Best” is a show I know about only from media references (and from the aforementioned movie “Pleasantville,” a critical parody of the show’s ethos). But that it would have been premised on the kind of culture described by the OP, which culture would have been highly retrograde by today’s standards, doesn’t surprise me at all. I’m certainly not a child of the 50s, having been born in 1979 (my memories stretch back to at least 1982). But years of acquiring factual knowledge have long taught me that the values of mainstream society were very different until mainly young people began to challenge them with the counterculture movement of the 1960s, which basically laid the foundations for the more tolerant, questioning, and egalitarian society of today.
There have been numerous related threads here; just recently we discussed girls often not being allowed to wear pants to school until the early 1970s. Or here is a classic thread that grew to be very long and that should be highly educational as to numerous strictures and expectations that society had well within living memory: In the dustbin of our cultural history.
Bland staple TV sitcoms and other media would have reflected established social expectations, not necessarily what everyone would have been happy to expect. Case in point: the OP describes how “Father Knows Best” performs mental hygiene encouraging girls and women not to aspire to having serious careers but to focus on getting a husband and becoming a homemaker. This was the “official line” at the time, to promote the ideal of a nuclear family where Dad was the sole breadwinner and Mom took care of the house, the kids, and her husband.
This was in fact a patriarchal reaction to what had happened in World War II, where tons of women took jobs to free up man to go into the Armed Forces. Women were massively employed in factories, some got jobs as pilots ferrying military aircraft across the Atlantic, others joined the women’s branches of the military (WACS, WAVES, WRENS, ATC, whatever), and supplemented the “men’s army” as clerks, drivers, postal workers, and so on. When the War ended, their conitnued employment was highly discouraged, as it was seen as keeping jobs from men who had served in the military and were now being demobilized. There was a lot of pressure for women to simply accept the idea that their proper role in society was to be housewives. Where this idea was not promoted by a woman’s friends, it was encouraged by gendered advertising, mainstream media, and the way the job market worked).
But this doesn’t mean that all women and blindly accepted such a fate. There is numerous data to the effect that many 1950s housewives were very dissatisfied with the role that society had foisted on them (for fictional examples, think of April Wheeler in the 1961 novel / 2008 movie “Revolutionary Road” and Laura Brown in the 2002 movie “The Hours”); and many of them turned to psychotherapy, alcohol, and prescription drugs in order to cope.
This dissatisfaction culminated in Betty Freidan’s landmark 1963 book “The Femnine Mystique,” in which the author talked about the lot of women in society and shared accounts of dissatisfaction of actual housewives. This book more or less singlehandedly kicked off Second Wave Feminism, during which women started to fight for having fully equal rights with men. To put things into perspective, First Wave Feminism (more or less the 19th century to the 1920s, suffragism, etc.) had achieved giving women the right to vote and hold public office, not to have their property taken and controlled by their husbands, not to have their physical person under direct control of their husbands, and to be able to get certain jobs. However, in the 1960s, it was still legal to pay women less than men, to withhold jobs from them on the basis of their sex, to require a husband’s or other male’s countersignature for credit cards, etc., and to demand sex of (i.e., rape) your wife. The status of hitting your wife was also a legally gray zone – though technically assault, the police were often reluctant to get involved in “private family conflicts.” It was finally during the 1960s and into the 1970s that these things were massively challenged by women and their allies.
To show just how socially acceptable crudely sexist media was as recently as the 1960s, look at this gem from 1967, an ad for Eastern Airlines solely featuring young women who are supposedly “the losers,” I.E. not good enough to be hired as stewardesses by them (immature, married, wears glasses, not conventionally feminine…) Furthermore, a print ad from the same “Presenting the Losers” campaign features a cluster of young models, including some of the most attractive ones of the time, like Ali McGraw. And that’s more than a decade after “Father Knows Best.”
But to get back to sitcoms: one that seems not to have been mentioned upthread is “The Dick Van Dyke Show,” which debuted in 1961. This show, while still showing the conventional family model of a father who brings home the bacon and a mother who is a homemaker, had a younger and in some ways slightly more progressive flair to it. Here is an interesting fact: The mother, Laura Petrie, wears slacks around the house. On one hand, this reflected real housewives at the time. On the other hand, it caused some controversy in production. IIRC, they wanted Moore’s character to wear house dresses all the time like Lucy Arnaz and June Cleaver, but Moore insisted on her slacks because, as already stated, real housewives were already wearing them at the time on a normal day at home.