Could a psychiatrist or psychotherapist run a clinic from home?

Yes, but few do. Mental health if pretty low-tech, so it does not require fancy equipment. Therapists still sometimes work out of their home, but many prefer the seapration of private and professional life that an office provides. If they want to work out of their home, they have to have a business license and follow city regulations--some of which Prohibit business in residential settings.
Agree w Dr. Elliot. I know psychoanalysts who see patients in a private office of their home. However, a big part of psychiatry/ psychotherapy is about learning to set appropriate boundaries. This might be more easily done when home and office are separate. This does not mean that treatment can not be provided from a mental health professional's home.