In solid state physics the electron density is often equated to $e|\psi|^2$. However, the Sakurai says (Chapter 2.4, Interpretation of the Wave Function, p. 101) that adopting such a view leads "to some bizarre consequences", and that Born's statistical interpretation of $|\psi|^2$ as a probability density is more satisfactory.
I am aware that a position measurement of an electron leads to (in the Copenhagen interpretation) a collapse of the wave function into a position eigenstate $x$ with the probability given by $|\psi(x)|^2$, and that it is known from some scattering experiments that the electron behaves as a point-like particle.
However, the electron density is experimentally observable, e.g. by X-ray scattering. One could argue that X-ray scattering is done with a large ensemble of atoms, so that what we actually observe is the average over many position measurements.
I am wondering if this the only "bizarre consequence", since this distinction seems to be a minor difference to me. My questions aims at clarifying the difference between the probabilistic interpretation of $|\psi(x)|^2$ and the interpretation as an electron density.