Is there folk knowledge associating late breastfeeding with low intelligence and/or gullibility?

I suspect it's just a poetic riff on the less well off in society.

I only did some cursory googling (searching for myths, misconceptions, etc.), but I failed to find the slightest trace of hints that breastfeeding until very late can lead to low development or anything of the sort. However, Lars's answer raises that late weaning was relatively common in less well off populations.

That being said, there was some crazy stuff out there in the late 19th century. For instance, Jacqueline Wolf, a historian of women's and childrens' health, notes in this interview on the history of the breastfeeding debate (webcache in case you disagree with Times' privacy policies) that:

There was a large group of doctors who feared that when girls were in school during the time they were going through puberty, their reproductive systems were competing with their brains for energy, and their brains were winning — the over-education of girls.

Also, there apparently are some (conflicting) studies on the topic of late leaning, with some of them apparently warning against it -- on the basis that it can introduce dietary deficiencies or increase the risk of getting food allergies. I've honestly no idea of what their merit are.

Yes.

Don’t Poison the Baby With Angry Breast Milk Several advice books suggested that mothers could harm their babies by thinking the wrong sorts of thoughts. The Sadlers, husband-and-wife doctors who collected their wisdom in 1916, blamed “angry” mothers for causing their babies’ colic. Mothers could also run dry by engaging in “worry, grief, or nagging,” they wrote. In his 1877 book, Advice to a Wife, Chavasse informed mothers not to nurse for too long. Once the baby was past 9 months of age, nursing could cause “brain disease” in babies and blindness in mothers.

It even gets worse. Much wore.

Breast feeding is nursing socialism!

A Spoiled Baby Is a Socialist Baby Before Spock’s 1946 book, a strict approach dominated baby advice books. Experts advised mothers to keep infants on schedules for feeding and sleeping. Holding them just for the sake of it was considered a sure way to produce what a 1911 text termed a “little tyrant.” As the U.S. Department of Labor observed in an “Infant Care” pamphlet in 1929, “a baby should learn that such habitual crying will only cause his parents to ignore him.”
_{Libby Copeland: "The Worst Advice Ever Given To Parents, Going Back To The 1700s", Slate, 11/27/2012 06:35pm ET // Tired of reading? Watch a video summary. (start at 2:00min)}

But: We have to look at the time this song was written and where.

It is indeed a simple allusion to backwardness and lower class. In other words: there was a certain folklore associating late breastfeeding with low intelligence and/or gullibility. (Old version of the question)
Or: there were and sometimes are cultural representations in various societies in history linking late breastfeeding with low intelligence. (Newer version of the question) Mainly past Western societies, although the battle over this practice is still ongoing. And now also in for example China, where imports and sales for milk powder and formulas keep on skyrocketing for some years now.

Whether or not to breast feed, how and for how long is as much a medical problem to solve, plus of course a basic human necessity, as it is and was a marker of social difference.

Still during the seventies women were often advised and believed it better to wean babies as soon as possible. Not doing that was seen as excessively traditionalist or hillbillyness.

From personal contacts I was told by mothers of that era that during the seventies all "modern women" were expected to wean their babies after 6 weeks.

This whole topic was traditionally associated with nursing mothers, vs wet nurses and weaning age from either, with vastly differing opinions. For what looks like mostly arbitrary 'reasons'.

That changed in 1867 when Henri Nestlé began marketing industrialised infant formula milk powder:

Though it is not known when Nestlé started working on his infant formula project, by 1867, Nestlé had produced a viable powdered milk product.(5) His interest is known to have been spurred by several factors. Although Nestlé and his wife were childless, they were aware of the high death rate among infants. Nestlé would have been aware of Justus von Liebig's work in developing an infant formula.(2) In addition, fresh milk was not always available in large towns, and women in higher society were starting to view breast feeding as an "unfashionable" option.

Ironically, these (early) infant formulas are clearly severely deficient in several nutrients essential for optimal development, but they were indeed the better option to letting the children starve or trying to feed them simply adult food way earlier than they could stomach it.
These formulas were improved as knowledge increased but have never been a perfect substitute, merely a sometimes adequate ersatz. With quite some scandals over their adequateness, production, distribution, marketing, use and health-effects, over the last 150 years.

By the 1950s, the predominant attitude to breastfeeding was that it was something practiced by the uneducated and those of lower classes. The practice was considered old-fashioned and "a little disgusting" for those who could not afford infant formula and discouraged by medical practitioners and media of the time. (WP)

That changed. Again. A few times, actually.

Breast milk is the optimal form of nutrition in infancy. Breastfeeding protects an infant from a wide array of infectious and noninfectious diseases. With very few exceptions, in the healthy term infant, breast milk alone (with vitamin D supplementation) meets all of the nutritional requirements up to six months of life. The Canadian Paediatric Society, Dietitians of Canada, and Health Canada recommend exclusive breastfeeding for at least the first four to six months of life, and continuing with complementary foods for up to two years and beyond (1). The World Health Organization recommends exclusive breastfeeding for the first six months of life in both developing and developed countries (2).[…]

That is our current, decidedly Western, common medical understanding of the issue of breast feeding and weaning. But this is a story with quite a ride in it, mainly during the late 19th and entire 20th century (or even earlier). The 'optimal' weaning time is still very much debated. This constant changing over and around of medical advice for this behaviour indicates quite strongly that it follows fashion rather than what all doctors and politicians (as we have to call them) thought of always at their respective time as science and scientific advice.

Durations of breastfeeding were generally longer in ancient times than in western society today. Aristotle stated that breastfeeding should continue for 12 to 18 months, or when menses restarted in the nursing mother. Mothers in Zulu societies have traditionally breastfed their infants until 12 to 18 months, at which point a new pregnancy would be anticipated. Ancient Hebrews completed weaning at about three years. Most children in traditional societies are completely weaned between two and four years of age.

Anthropological theories have recommended final weaning at the following points: when the infant acquires four times his birth weight; when the infant’s age is six times the length of gestation (ie, 4.5 years); or when the first molar erupts (6).

The inappropriate early introduction of mixed feedings began in early 19th century western society. Prominent physicians at that time, such as American Pediatric Society founders Dr Luther Emmett Holt and Dr Job Lewis Smith, recommended that weaning begin at around nine to 12 months of age, or when the canine teeth appeared (6). Smith recommended against weaning during the summer months because of the risk of ‘weanling diarrhea’. Unfortunately, as weaning began earlier and earlier in the 19th century, infant mortality increased. Introduction of weaning foods was an important cause of infant mortality in the 19th century. This increase in infant mortality, in part, spurred the development of paediatrics as a specialty in medicine (6).

In the early 20th century, mothers were encouraged by the medical community to raise their children scientifically or ‘by the book’. In the 1920s, the United States government published Infant Care, which at the time was referred to as the ‘good book’ and was read by women from all socioeconomic statuses. It recommended cod liver oil, orange juice and artificial feeding.

By 1940, the Honourable Paul Martin, Minister of National Health and Welfare, Ottawa, Ontario, published The Canadian Mother and Child (8th edn, 1949) written by Dr Ernest Couture. Over two million copies were distributed to new and expectant mothers before its first revision in 1949. Couture emphasized breast milk as the ideal form of nutrition for babies.

More recently, according to Health Canada, in 1998/1999, 81.9% of children were breastfed for some time. Among those infants who were breastfed, 63.0% were still breastfeeding after three months. Breastfeeding duration rates vary depending on maternal age. While only 49.1% of breastfed infants of mothers 25 years or younger continue to breastfeed after three months, 74.9% of breastfed infants of mothers 35 years or older continue to breastfeed beyond three months (7). The most common reason mothers give for weaning is a perceived insufficient milk supply. Among women who breastfeed for longer than three months, one of the most important reasons for weaning is returning to work (8).

In the United States, rates of breastfeeding are lower. As of 1998, 64% of infants are breastfeeding at hospital discharge and 29% are still breastfeeding at six months (9).
_{Canadian Paediatric Society: "Weaning from the breast", Paediatr Child Health. 2004 Apr; 9(4): 249–253.}

We see that while earlier the better off elites of society had a much higher chance of being breastfed by wet-nurses, and perhaps even their own mothers, with the rise of milk powder and formulas the better offs tended to be fed 'by the books', that is: weaned early.

These methods of feeding are not only 'modern' but also expensive. This leads to a very simple observation: This is not a musical folklore but an allusion to medical folklore! And social! Breast feeding and weaning time was and is judged according to the fashions. Now, we 'know' that not breast-feeding at all is 'bad' but and weaning too early is bad as well. But we do not know the 'ideal' time to wean. But that never stopped the gurus to hand out guidelines – which are then followed more strictly by the upper classes, and then those tend to fare better in life, mainly by fiscal inheritance, but this produces a spurious correlation with intelligence; ie here: breast feeding morals.

For the US specifically, the differences were and are sharpest between 'whites', 'blacks' and 'hispanic':

Turnaround in the 1970s
[…] The overall downward trend in breastfeeding incidence reached its nadir at 22% in 1972 (Hendershot, 1981) (Figure 3-1). The subsequent increase was not uniform across the population. Breastfeeding incidence rose among white and black mothers, although the increase was greater among whites (Hendershot, 1981). Although data on Hispanic mothers are incomplete, there are indications that their breastfeeding incidence remained stable or even continued to fall (Smith et al., 1982).
_{"Who Breastfeeds in the United States?", in: Nutrition During Lactation. Institute of Medicine (US) Committee on Nutritional Status During Pregnancy and Lactation. Washington (DC): National Academies Press (US); 1991. (link)}

And this continues systematically

Hispanic cultural and familial expectations to breastfeed; and (3) Adapting to life in the United States: cultural norms in conflict. Women said they were motivated to breastfeed because of their knowledge and observations of its health benefits for mother and child. They said breastfeeding is ingrained in their Hispanic cultural heritage, and infant feeding choices of female family members were particularly influential in women’s own decision to breastfeed. Women said they experienced embarrassment about breastfeeding in the United States and as a result, often chose to initiate formula feeding as a complement so as to avoid feelings of shame. Additionally, they faced economic pressure to work, key barriers for continued breastfeeding among Hispanics in the United States.
_{Sarah Hohl et al.: "Cultural Norms in Conflict: Breastfeeding Among Hispanic Immigrants in Rural Washington State", Matern Child Health J. 2016 Jul; 20(7): 1549–1557. DOI}

The chaotic pendulum circles over different centers at different times:

Nutritional deficiencies with increased morbidity (e.g., scurvy and rickets) appeared as breastfeeding diminished. Data from the US census of 1900-1910 revealed that children who were breastfed had a 40% lower mortality rate than did their formula-fed peers. Modern medicine adopted calorimetric methods for infant feeding and the concepts of bacteriology to help advance a ”clean milk campaign” that favored the advocacy of formula feeding. By 1950, pediatricians recommended the introduction of vegetables to the diet at age 4 months. Thus, the advent of commercial interests and the modern professional advice led to the decrease in the rate of breastfeeding in the twentieth century.
_{Yvette Piovanetti: "Breastfeeding Beyond 12 Months. An Historical Perspective", Pediatric Clinics Of North America, Volume 48 * Number 1 February 2001. DOI}

Ironically, at the same time women in some deprived areas of the world are abandoning breastfeeding at an alarming rate because they believe it to be "low class" and "illiterate."
_{Marie Walter: "The Folklore Of Breastfeeding", Bul. N.Y. Acad. Med., Vol.51, No.7, July-August 1975.}

Some back and forth in social aspects of what was seen as purely scientific or at least rational judgement:

This was the situation in the Workman home, where the wet nurse, dubbed "Irish Mary," retained her position in spite of serious problems. According to Workman, Mary never learned how to care for her suckling and, on one occasion, put the child in mortal danger. Workman had observed her standing in the middle of the street, holding the baby, and staring at a rapidly oncoming carriage. Workman's complaint about Mary's lack of knowledge echoed that of many physicians. Conflating immorality with ignorance, pediatrician Rowland Godfrey Freeman alleged that "unmarried mothers are women of a low grade of intelligences," and, as a result, they "cannot be trusted to care for the baby on account of ignorance or unreliability. "

A week after Mary's departure Workman found a replacement, a wet nurse she never referred to by name, but who might well have been called Irish Mary the Second. The woman resembled her predecessor in a number of ways, including her background, her adjudged lack of intelligence, and her concern for her own baby. The new employee claimed to be an Englishwoman but soon revealed her true origins when she opened her mouth and spoke in "broad Irish." Although the wet nurse was healthy, Workman found her appearance unattractive and referred to her face as having a "most heavy, unintelligent mould." Consequently, the wet nurse required constant observation.

What concerned Louise J. were obvious defects, not hidden diseases. She suspected that wet nurses lacked even "average mental or moral qualifications"; and the response she received supported her assumption. Wet nurses did not come from "the highly-intelligent classes" the expert attested, but added that few were "distinctly vicious." They were, however, women who had given birth out of wedlock. The reference to this fact by Louise J., who admitted at the outset that she intended to use wet nurses in the future, suggested that morality remained an intractable complaint - one that did not prevent a wet nurse from being hired, but one that also never ceased to cause concern. Like the drone of employers who complained about their lazy servants but could not live without them, the frequent references to the moral limits of wet nurses were overshadowed by the reality of their effectiveness. Both Louise J. and her respondent knew this.

The most active forum for scientific mothers, Babyhood magazine, was "Devoted Exclusively to the Care of Infants and Young Children, and the General Interests of the Nursery." It began publication in 1884, supplying expert advice to women who viewed child rearing as a highly demanding discipline. The magazine employed leading pediatricians to write articles and answer questions from readers. By 1885 it had earned an endors*m*nt from the American Medical Association. One of its editors, Leroy M. Yale, was a physician and at one time, a lecturer on diseases of children at Bellevue Hospital Medical College in New York City. The other editor was the domestic writer Mary Terhune, who had denounced wet nurses in her book Eve's Daughters. The prolific author of twenty-five popular books on domestic matters, as well as twenty-five novels, Terhune (who published under the name Marion Harland) believed strongly in artificial feeding. Not unexpectedly, when subscribers opened the pages of Babyhood or similar publications, they found a plethora of advertis*m*nts for commercially manufactured infant foods bordering the many articles about infant feeding. Terhune herself penned the advertising booklet for Carnrick's Soluble Food. (p 162–165)

The founding in 1956 of the La Leche League, an organization dedicated to "the womanly art of breast feeding," and the resurgence of maternal nursing in the "baby bust" years that soon followed, suggest that a fundamental reconception of middle-class maternity was underway. According to the National Center for Health Statistics, the number of American women nursing their babies immediately after birth rose from 24.7 percent in 1971 to 55.3 percent in 1981. In many cases the duration of nursing proved very brief. Even so, the figures seem to indicate that women believed they should make an effort to breast-feed. In. 1982 maternal breast-feeding rates peaked at 61.9 percent, although by six months of age only 27.1 percent of babies were still being breast-fed. The latter figure was, nevertheless, five times higher than it had been in 1971, when the rate was only 5.4 percent.

The reasons for the recrudescence of maternal nursing are numerous and to some degree contradictory. They include a feminist and countercultural rejection of medical authoritarianism (also evident in the embrace of the natural-childbirth movement) and at the same time reflect the continuing influence of the earlier "baby boom" generation, which constructed motherhood as a vocation. Yet, motherhood was not women's only vocation in the 1970s. The growth in maternal nursing paralleled an increase in the labor force participation rate of women with young children. This points to another critical factor: growing rates of maternal nursing were not universal, but were class based. Middle-class mothers embraced breastfeeding at a time when low-income women continued the practice of bottle-feeding. The use of mothering style as a demarcation of class — evident in the nineteenth century when well-to-do women saw breast-feeding by lower-class women as animal like – obviously continued in the late twentieth century.
_{Janet Golden: "A social history of wet nursing in America. From breast to bottle", Cambridge History of Medicine, Cambridge University Press: Cambridge, New York, 1996, p204. DOI}

It should be self-evident from the above (and really the question here in itself) that the song line didn't age so well, as the current fashion would suggest the opposite: longer feeding yields better outcomes in health in later life.
So what remains from the question of "a late breastfed fool" is what looked like a supposedly scientific bit of knowledge filtered down into common sense and folk wisdom is really nothing more than a derogatory epitheton with exclusively cultural and class-based explanatory value.

the body in linguistic-anthropological analysis, aiming to account for “the fact that the social agent is before anything else a being of flesh, nerves, and senses [...] a ‘suffering being’” (Wacquant 2004:vii; see also Wacquant 2015). This approach demands attention to the question of what it means to have, or be, a body. According to Merleau-Ponty (1962:111), “having a body means being united with a definite milieu.” This phenomenological presumption bears resonance not least with regard to the bodily practice of breastfeeding, a practice that is obviously deeply gendered, situating the eminently female body in a moral and existential nexus of images of motherhood, childcare, and health. Thus, as Beauvoir (2014) points out, the body is not only in a situation, it is a situation (cf. Moi 2001:59– 67). This is a reason for thinking about the workings of discourse in terms of the notion of the historical body—that is, an embodiment of discourse, events, and actions in which the individual has engaged. To say that the body is a situation is to acknowledge that the body that is located in a situation or a “definite milieu” also has a history, constituting its own premises. However, because the body continuously changes, it also has a contemporary existence, which may not be in harmony with the ideas that the historical body produces about how one’s body really is or should be. In this sense, to experience oneself as a failing body originates in an incongruity between the historical body and the body that is manifest in the present.
_{Linnea Hanell: "The Failing Body: Narratives of Breastfeeding Troubles and Shame", Journal of Linguistic Anthropology, Vol. 27, Issue 2, pp. 232–251, 2017. DOI}

This 'discourse' or the physical realities seem to be an anthropological constant, though:

Comparison of the results from the four stable isotope studies to those of other published studies reveals that the modal age at the end of weaning was slightly lower in agricultural communities than hunter-gatherer communities, but the range of ages was similar. Weaning prior to the age of eighteen months was rare before the post-medieval period. It is argued that the gradual reduction in breastfeeding duration since the Neolithic, and the replacement of breastmilk with animal milk products, means that on the whole the development of agriculture probably served to increase infant morbidity and mortality.
_{Rachel Howcroft: "Weaned Upon A Time Studies of the infant diet in prehistory", Dissertation, Stockholm University, 2013. (PDF)}

But maybe this line from the song can still resonate in certain demographics or even more in mental pictures others have about them:

_{R J Harris: "Nutrition in the 21st century: what is going wrong", Arch Dis Child 2004;89:154–158. doi: 10.1136/adc.2003.019703 (PDF)}