Homeopathy. does. not. work.
Yes it does. Psychosomatic illnesses are real illnesses, and very common, and can be cured by appropriate treatment.
But the medical profession will just say "it's all in the mind" and leave you to die.
Yes, it's all in the mind, but you still need help. If that means being conned, well, the mind's like that. If it's not kept conned, it loses the will to live.
Placebos are especially appropriate for children who are made ill by their mothers. They get better when their mothers expect them to.
This may be why acupuncture works on dogs. I bet it wouldn't work on cats.