The problem with Ensemble interpretation is that the quantum state of a uranium atom in its ground state, including the nucleus and the electrons, is a property of one atom, not of an ensemble. You don't need to measure the electron position, you can just subject the atom to external forces, and see how it moves. This motion is predicted by the wavefunction of the one atom, not of any ensemble.
It is logically possible to reject this position, of course, but then you are claiming that a quantum description cannot be given to an individual system, and you renounce the ability to map physical objects to mathematical objects so that the behavior of one is isomorphic to the behavior of the other. This is an enormous weakening of the goal of physical science, that there would be no point in doing physics. Also, it throws away the many cases where quantum mechanics makes firm predictions about the behavior of individual systems.
The Wikipedia page erronously claims that Einstein supported the Ensemble idea. This is false, but there is a quote that makes it seem true. Einstein was referring to hidden variables here, and the reason he states it in terms of ensembles is because he believed that the wavefunction described a statistical pattern for hidden variables which is analogous to statistical mechanical distributions. These statistical distributions do not describe a single particle (at least not in a straightforward interpretation), because the particle has a hidden position deep down underneath. The averages of many position measurements require an ensemble, and these measure the wavefunction. Einstein used the term "ensemble" in the context of quantum mechanics to emphasize the probabilistic nature of the wavefunction, not the inability to describe an individual system.
Einstein also believed that the quantum mechanical description of an individual system, inasmuch as it was deterministic, was correct. It was only the probabilistic aspects of the description that required hidden variables, of course, just like classical theormodynamics. You don't need an ensemble to describe the motion of piston pushing a gas, only for the statistically random motion of individual molecules.
A true ensemble interpretation renounces the description of a physical system entirely, leaving only the description of the statistics of identical measurment. It is not what Einstein had in mind, and it is not reasonable in the majority of cases where quantum systems are in their ground state, and changing adiabatically, so that they essentially reproduce deterministic classical behavior.