As far as I can tell the Copenhagen Interpretation basically is a shut-up-and-calculate interpretation. It's an operational theory that is only capable of predicting subjective-ish experimental results, and doesn't make claims about the "contents of reality". That is to say, all its predictions are of the form "if I did [EXPERIMENT] I would observe a result according to [DISTRIBUTION]". Which is somewhat respectable (although what exactly counts as an observation is naturally ill-defined, since the theory doesn't encompass the observer itself).

The real problem is that having the CI as the majority view sucks people into philosophical positions where you're not allowed to even wonder what reality is made of, or how these observations are manifested. See: "the EPR experiment proved there's no such thing as reality, right?"