Why would people do something that might be Avoda Zara just for the placebo effect?
Who says it doesn’t work? I once saw somewhere (I don’t remember where) that the ta’ava for a”z was that it worked – through koach hatumah. Not ch”v a separate r’shus, but rather allowed by Hashem to to give us s’char for following His will even if it means foregoing the benefits.