The answer, as for many things in life, is "It depends." If cost is not a consideration, it can sometimes still be "It depends."
Here are some factors to consider:
Many people (even without memory issues) strenuously resist the idea of "strangers" in their home, repeatedly refusing home care, firing workers, even being abusive to them or family members because they are there.

Isolation at home - even with caregiver(s) present can make dementia progress faster. Socialization at an appropriate (and changing) level in assisted living can help with physical and emotional well-being.

Stress in the home if the loved one is disruptive to home life can make a very tense atmosphere for all, especially anyone who cannot understand the changes and demands of living with a person with dementia, like a spouse with their own needs or young children. Also, some homes are just too busy to be suitable for keeping a person with dementia on an even keel.

Even with absolutely stellar home health care, living with a person with dementia is extremely stressful to the spouse or other primary caregiver in the home. They always feel responsible for overseeing and augmenting care, which is exhausting.

If the person lives alone, paid health care workers are hard to retain so continuity of care can be very spotty with many different caregivers in the course of a week. The there is the issue of who monitors what is going on day-to-day.

All that said, some people have the personality, patience and time to care for their loved ones at home, sometimes with help from family. But even they face stress, isolation and exhaustion doing it, especially long-term and with family members in the background offering advice but no hands-on assistance.

Are you asking as a someone making this decision about a loved one for whom you are caring?