...it doesn't really tell you that, say, "Openness" is a valid physical construct in people's brains...

I think this all depends on what your model predicts and with what accuracy. To use an analogy, a thing like "temperature" doesn't really exist -- it's just an illusion caused by the motion of particles -- but it's still a very useful concept. In most cases, we can close our eyes and pretend that "temperature" is a measurement that really does refer to some physical quantity out in the real world.

So, is "Openness" (or any other of the Big Five axes) like "Temperature" ? Can we even measure a person's "Openness" value reliably ? If so, what does it tell us ? What verifiable predictions can we make based on it ?

In the case of Myers-Briggs, AFAIK, the answers are "no", "nothing much", and "none". In the case of temperature, we can definitely answer "yes" to the first question, and list a whole bunch of things like melting points and specific heat values etc. in response to the other two.