Why do some medical professionals and other caregivers talk to the elderly as though they are small children, even when their mental faculties are intact?

The type of “baby talk” you’re referring to is typically reserved for young children. Still, sometimes it creeps into dialogue with people with disabilities or older people, particularly those with cognitive impairments. While it may not be on purpose, the “infantilization” reflects ageism at its core.
“Baby talk” directed to seniors is usually rooted in a perception that eldercare is equal to child care. In some cases, caregivers even adopt infantilized language based on the perceived level of incompetence of the senior. This kind of “baby talk” is not well received. Older adults prefer to be addressed like adults.