This understates the capacity for anti-depressants to reduce depression, because placebo treatment is also better than no treatment, and if one antidepressant doesn't work, you can try another one.

I'm not really sure whether that the case. Don't work often doesn't mean that it shows no effect. If you eat a vegetarian diet you will feel different. Different enough that all sorts of people think that becoming a vegetarian will cure all your problems.

Deciding whether the kind of mental state that you get through a anti-depressent is really better is a nontrival task.