While it was a shock for myself as well, it is true that such customs existed and was happening frequently even until recent times. But was it the main reason for marriage just as Smith said, not at all, and was any ideologic belief behind this custom, hell no.

Beforehand, the quote you mentioned is a part of 'The wealth of nations', written in 1766. With that in mind, I will focus on the era of Qing(1644-1912) in history of China, as it shares the days that Adam Smith lived in(1723-1790). Also, as it deals with children, I will speak not only about infanticide but the traffiking of a child as well; it will help understand the status of a girl in the culture of China.

For the first part, the sale of a child. 童養媳(tong yang xi) is a word composed of child(tong) adopted(yang) xi(bride), meaning 'a girl adopted to become a future bride of a boy'. And in this custom, money were exchanged in turn for handing over the bride. And the Qing jurisdiction system forgave such deals over the childs faith, while not confirming it as a form of marriage; thus it was half-official for poor houses to sell their daughters, for the poor to reduce money spent for raising a child & profit from the sale, and for the buyers to assure they have a future bride which was treated as their 'property'. Of course, there existed crimes such as kidnapping and human traffiking, but those were all illegal; and were sentenced hardly under jurisdiction.

For the infanticide, there exists another notorious word for it;溺女(ni nü), as drowning(ni) girl(nü). And this was a custom which had a gruesome history; the common method was to drown the child by pushing her head into the pot filled with water until she suffocate. Another way of infanticide was a simple exposure; and this custom was done in urban areas more often, as it was easy to remain anonymous. Yes, these acts were of course illegal, and yes, Qing officials recognized the situation. They built orphanages for abandoned children; countless officials petitioned for the policy to abolish such inhumane acts; and even the emperors themselves ordered decrees and specific ordering for child support; famous poets also wrote poems criticising the cruelty of infanticide. In one specific case, although this was in 1838, the Lieutanent governor Ke of Guandong province promised 'retribution of heaven' for infanticide and did act accordingly, while writing essays condemning the customs. It is mentioned that Ke did took success, but for the question of success for other policies, I do not know; but certainly such cases existed even after all that works done by emperors and governors, until the collapse of Qing and rise of PRC, and led onward until even modern days.

The reason for such customs is a diverse topic, as it includes many key factors: namely

1. Extreme poverty of lower class, especially in the rural areas; 'unprofitable' physical status of female in labor; 'family size control'
2. The astrological belief of luck and disaster depending on the date of birth of a child
3. Confusianic beliefs which surely played a role in downplaying female social roles compared to male
4. Lack of moral conception on a newborn life And so on. Each scholars cite different aspects from these or from something else, and there is no true answer for this question, 'Why did it happen?' But we need to note that Confusianic belief NEVER encouraged infanticide or any kind of structural discrimination on sexes; one of five main dogma which is central in eastern philosophy is 父子有親(ba zi you qin), father and child should be close and in freindly relation.

To sum up, the horrible deeds which Smith wrote, was indeed a custom for Chinese, even for very modern era. And there are records that catholic missionaries who headed for China were shocked with sight of exposed infants, so probably Smith got the story from them; as he never once laid foot upon China.

If you wish to see specific examples for each case, please refer to the books and articles below. Actions of Qing officials are listed well in Mungello and Jimmersons work, and legal basis with explanation of Tong Yang Xi in Huangs.

Reference:

• Drowning girls in China: Female infanticide in China since 1650, D. E. Mungello

• The history and theory of legal practice in China: Toward a historical-social jurisprudence, Phillip C. C. Huang & Kathryn Bernhardt

• Female infanticide in China: An examination of cultural and legal norms, Julie Jimmerson

• Female infanticide in China, Bernice J. Lee

• Family size control by infanticide in the great agrarian societies in Asia, John C. Caldwell & Bruce K. Caldwell