The cultural (rather than medical) reasons why nighttime separation is encouraged in the United States stem from the Puritan tradition of independence and individualism. A negative reaction to breastfeeding in public is often attributable to another Puritan attribute, that of modesty.
The cultures in which children share a bed with their parents, such as those South Asia and Southern Europe, tend to be more family-oriented and less individualist than the United States. In these societies, there is less emphasis on gaining independence from one's parents and establishing one's own life as quickly as possible. As the attached article points out, there have been medical claims that sharing a bed can be harmful to children. The American Academy of Pediatrics has linked the practice to Sudden Infant Death Syndrome, for instance. However, the cultural practice was established long before such clinical justifications for it arose.
Breastfeeding in public is a related issue in the sense that Puritan culture prescribes both independence and modesty. Most objections to public breastfeeding are centered around the idea that women should keep their breasts covered. Someone with strongly Puritan convictions would probably object to both bed-sharing and public breastfeeding. However, another person who cared mainly about individualism or felt that the utility of public breastfeeding was more important than strict adherence to modesty would condemn the former but not the latter.