They sometimes do work, because of the placebo effect.
For example look at https://en.wikipedia.org/wiki/Hospitalism - infants will themself to death because they are not loved.
Ask doctors and nurses and they'll tell you about people with and without the will to live, and how having a good outlook can mean the difference between life and death.
These are infants who are well cared for, fed, clean, warm. But they die simply because they are not held.
Even if they did die is quite a stretch to assume they willed themselves dead. Infant lack a mental concept of death or dying to even be able to will themselves dead.
As for a cause, I would assume something similar to amblyopia.
So it cured someone, yet didn't work?
I don't think you realize how powerful the placebo effect actually is.
But that is simply incorrect, placebos can cure, yes, actually cure. And not just mental conditions either (which is another common, yet incorrect, belief).
Last night the baby went to sleep because I walked counter-clockwise while singing lullabies? That's it, that's the only direction I can walk in from here on out while putting her to bed. The kid ate her veggies with no complaint because I was humming The Simpsons theme? It's now the official meal-time song.
When she gets older and starts having well-developed preferences, then I will no longer be crazy - only my behavior will be, because I'll be in thrall to a tiny crazy person.
Talking to nurses working with babies apparently this is a thing and they get to understand babies quite well when they cry.
Parents rate their children as hyperactive after consuming sugar, no matter if the "sugar" is actually just red colored water. It's perception that's changed.