I think much of this simply comes down to the fact that the "science" of Psychology is highly unscientific most of the time. Even when psychologists do studies in a scientific format the sample size is usually too small to have any statistical validity.
It's very easy to do simple scientific studies on this stuff and determine it isn't "true" but then it was never meant to be - they're thinking tools, models intended to increase understanding and hopefully do better than pure chance. Meyers-Briggs and the some of the various coarse generalisations in NLP, although easily falsifiable, do have some correlation with reality. NLP actually scores points here for having a blatant rip-off of Meyers-Briggs (in the form of "Meta-programs") that you don't have to spend vast sums of money on.
I completely agree that it's lazy and probably ethically dubious to use these kinds of extremely inaccurate tools to determine anything about employees careers but unless managements care enough about their employees to take the time to understand their motivations and preferences as individuals then I expect they'll keep using the latest fashionable pseudo-science.
Psychology is complex! Your placebo example for headaches is probably not as representative as it should be since the causes are often fairly well understood. In psychiatric conditions and many physical ones where "stress" is a major factor, the placebo effect is typically in the range of 30-50% and psychiatric drugs often only beat placebo by a few percent (if indeed they do at all, as e.g. many anti-depressants seem to have been approved largely through publication bias). For many forms of psychological intervention its incredibly difficult to design the equivalent of a double-blind placebo controlled trial.
The problem is not the current management fads but that we presume to be able to systematically determine who's going to do well in a certain job, or be motivated by a particular management style. So, while you hate these things for their lack of scientific grounding, I doubt we're even close to credible alternatives (and I'd love to be proven wrong, so if you've got some examples of methods companies should be using instead please tell). We make this whole situation worse by having employment laws that scare managers away from doing anything that could be considered biased towards one individual over another. This pushes HR departments towards some kind of standardised process such that everyone can be seen to be treated exactly the same and thus into the arms of the psychological pseudo-scientists.