Well, one more thing we have to take into consideration is that many illnesses are self-limiting, which means that one would eventually get better without any intervention whatsoever (common cold and a lot of other viral illnesses are a good example here). But because we are humans and our minds are prone to search for patterns, we would probably attribute the recovery to the remedy that we were taking at the moment.
Take the common cold (which I'm suffering from at the moment). When I was younger and had a cold, I used to take vitamin C and eat a lot of garlic (these are two most popular home remedies over here). And naturally I got better, so I thought it worked. Having read more about it, I realised that common cold is self-limiting, so now I just try to take things easy for a couple of days and drink plenty of liquids (hydration is important if you have a fever). And my colds go away as quickly as before. Naturally, I lack the comfort measure that vitamin C used to provide, namely "I'm actively doing something to get better, instead of waiting for the illness to resolve on its own." But since I don't believe in it anymore, there's no sense in taking it, even as a placebo.